{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:RFGC63ELN5KOG2XVSQ3HO6SRY4","short_pith_number":"pith:RFGC63EL","canonical_record":{"source":{"id":"1011.3334","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-15T10:34:55Z","cross_cats_sorted":[],"title_canon_sha256":"5e72354fe59d060295273c10fe5dcdb404b4cae71bbb0559478c25901011448d","abstract_canon_sha256":"48df557fba0d1a4a8f015dc588ab6fc860f91a6840c3d0f464f912ed7b4e732f"},"schema_version":"1.0"},"canonical_sha256":"894c2f6c8b6f54e36af59436777a51c70a6b32c166207775f7f8033eccdde978","source":{"kind":"arxiv","id":"1011.3334","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.3334","created_at":"2026-05-18T04:35:56Z"},{"alias_kind":"arxiv_version","alias_value":"1011.3334v1","created_at":"2026-05-18T04:35:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.3334","created_at":"2026-05-18T04:35:56Z"},{"alias_kind":"pith_short_12","alias_value":"RFGC63ELN5KO","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RFGC63ELN5KOG2XV","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RFGC63EL","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:RFGC63ELN5KOG2XVSQ3HO6SRY4","target":"record","payload":{"canonical_record":{"source":{"id":"1011.3334","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-15T10:34:55Z","cross_cats_sorted":[],"title_canon_sha256":"5e72354fe59d060295273c10fe5dcdb404b4cae71bbb0559478c25901011448d","abstract_canon_sha256":"48df557fba0d1a4a8f015dc588ab6fc860f91a6840c3d0f464f912ed7b4e732f"},"schema_version":"1.0"},"canonical_sha256":"894c2f6c8b6f54e36af59436777a51c70a6b32c166207775f7f8033eccdde978","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:35:56.114382Z","signature_b64":"JTjNbSwJJpjpwR6g/IAU60HNYJARYP9v3IEJwFzeQyLIccOMf7aC76XRV/DqKhuiHaCx9FTH7pEf1UYpdp9yAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"894c2f6c8b6f54e36af59436777a51c70a6b32c166207775f7f8033eccdde978","last_reissued_at":"2026-05-18T04:35:56.113920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:35:56.113920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.3334","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":"2026-05-18T04:35:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XnG3RZ8DbUJwHsLR4Havb1lrSgm5IvyJSn8fz7pHyII1+2X5L8AxhnuyPAfjheW3mjYXvTP0h0e5/GJD0p0SAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T05:22:05.319951Z"},"content_sha256":"3e0d21a5234c193cfbac8dac83a5017a4940fedaf5f2932eee509a7b2d271c2d","schema_version":"1.0","event_id":"sha256:3e0d21a5234c193cfbac8dac83a5017a4940fedaf5f2932eee509a7b2d271c2d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:RFGC63ELN5KOG2XVSQ3HO6SRY4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christoph Walker","submitted_at":"2010-11-15T10:34:55Z","abstract_excerpt":"The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator-prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3334","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":"2026-05-18T04:35:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FDLbhBLJNTEjDac9rzvek2AyCIEmZtRf9L3jDW9lwbe/xruRCJBtzQEBVjbU2rHYh9DDX2qnutCdtsQzk8Y9Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T05:22:05.320311Z"},"content_sha256":"a64e1ee9644fa9d6f52360d104e57f33deed9b6c920687c74d41b84983cbe551","schema_version":"1.0","event_id":"sha256:a64e1ee9644fa9d6f52360d104e57f33deed9b6c920687c74d41b84983cbe551"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4/bundle.json","state_url":"https://pith.science/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4/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":"2026-06-29T05:22:05Z","links":{"resolver":"https://pith.science/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4","bundle":"https://pith.science/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4/bundle.json","state":"https://pith.science/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RFGC63ELN5KOG2XVSQ3HO6SRY4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:RFGC63ELN5KOG2XVSQ3HO6SRY4","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":"48df557fba0d1a4a8f015dc588ab6fc860f91a6840c3d0f464f912ed7b4e732f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-15T10:34:55Z","title_canon_sha256":"5e72354fe59d060295273c10fe5dcdb404b4cae71bbb0559478c25901011448d"},"schema_version":"1.0","source":{"id":"1011.3334","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.3334","created_at":"2026-05-18T04:35:56Z"},{"alias_kind":"arxiv_version","alias_value":"1011.3334v1","created_at":"2026-05-18T04:35:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.3334","created_at":"2026-05-18T04:35:56Z"},{"alias_kind":"pith_short_12","alias_value":"RFGC63ELN5KO","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RFGC63ELN5KOG2XV","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RFGC63EL","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:a64e1ee9644fa9d6f52360d104e57f33deed9b6c920687c74d41b84983cbe551","target":"graph","created_at":"2026-05-18T04:35:56Z","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":"The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator-prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived.","authors_text":"Christoph Walker","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-15T10:34:55Z","title":"Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3334","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":"sha256:3e0d21a5234c193cfbac8dac83a5017a4940fedaf5f2932eee509a7b2d271c2d","target":"record","created_at":"2026-05-18T04:35:56Z","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":"48df557fba0d1a4a8f015dc588ab6fc860f91a6840c3d0f464f912ed7b4e732f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-15T10:34:55Z","title_canon_sha256":"5e72354fe59d060295273c10fe5dcdb404b4cae71bbb0559478c25901011448d"},"schema_version":"1.0","source":{"id":"1011.3334","kind":"arxiv","version":1}},"canonical_sha256":"894c2f6c8b6f54e36af59436777a51c70a6b32c166207775f7f8033eccdde978","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"894c2f6c8b6f54e36af59436777a51c70a6b32c166207775f7f8033eccdde978","first_computed_at":"2026-05-18T04:35:56.113920Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:35:56.113920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JTjNbSwJJpjpwR6g/IAU60HNYJARYP9v3IEJwFzeQyLIccOMf7aC76XRV/DqKhuiHaCx9FTH7pEf1UYpdp9yAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:35:56.114382Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.3334","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e0d21a5234c193cfbac8dac83a5017a4940fedaf5f2932eee509a7b2d271c2d","sha256:a64e1ee9644fa9d6f52360d104e57f33deed9b6c920687c74d41b84983cbe551"],"state_sha256":"938326dfaec70e83b3641c3962fe484a058f9822fc870f5217481ab0aeb3c884"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H6GiAmquyE1ycg/OV9Nt/9o28yP7h5mUIpLrg6xw2xFCkGEW9e8yNdWg1FqeepIHevX4tXFCooD0vrG5tcRDBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T05:22:05.322382Z","bundle_sha256":"c57d4b6233481167364e356ddc742affad66f56be29768bdf48b580f3af63be3"}}