{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:DVEKBN64BKRT3D4X7VGY5MVPRL","short_pith_number":"pith:DVEKBN64","canonical_record":{"source":{"id":"1907.05821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-12T16:20:26Z","cross_cats_sorted":[],"title_canon_sha256":"6748d4a18093d9cb7ea87666f5b86fa6ab35ac0207ed0d3d14c250c321b804f2","abstract_canon_sha256":"2c18d572c6a6a1137b73da08e1e488f83b08df6f47349e5111697a60dbbb80ad"},"schema_version":"1.0"},"canonical_sha256":"1d48a0b7dc0aa33d8f97fd4d8eb2af8ac4ed567960a1b83837390671c28c22f5","source":{"kind":"arxiv","id":"1907.05821","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.05821","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"arxiv_version","alias_value":"1907.05821v1","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.05821","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"pith_short_12","alias_value":"DVEKBN64BKRT","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"DVEKBN64BKRT3D4X","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"DVEKBN64","created_at":"2026-05-18T12:33:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:DVEKBN64BKRT3D4X7VGY5MVPRL","target":"record","payload":{"canonical_record":{"source":{"id":"1907.05821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-12T16:20:26Z","cross_cats_sorted":[],"title_canon_sha256":"6748d4a18093d9cb7ea87666f5b86fa6ab35ac0207ed0d3d14c250c321b804f2","abstract_canon_sha256":"2c18d572c6a6a1137b73da08e1e488f83b08df6f47349e5111697a60dbbb80ad"},"schema_version":"1.0"},"canonical_sha256":"1d48a0b7dc0aa33d8f97fd4d8eb2af8ac4ed567960a1b83837390671c28c22f5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:44.305262Z","signature_b64":"e4nvRayXYLHyR7Ha/P/bww1HdUp+nU2d51ZkuKWAf+IZZ+275JMb79rPVyqo8vhpiQ82v/Uuxg0qRhV6u4mPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d48a0b7dc0aa33d8f97fd4d8eb2af8ac4ed567960a1b83837390671c28c22f5","last_reissued_at":"2026-05-17T23:40:44.304701Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:44.304701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.05821","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":"2026-05-17T23:40:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wqMOSpskA/sTrs4J11SpnfG15ueYTyM4n5OLZU1u2FFgcve6mWUMayNMQMWlAr3TxM7AIDa5+Ct4EwtcG1ctDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:24:40.737416Z"},"content_sha256":"7f749af90f0604c27d2c4a0166a88b31e3e8d349d36d2dc6179b90ba3f6b4662","schema_version":"1.0","event_id":"sha256:7f749af90f0604c27d2c4a0166a88b31e3e8d349d36d2dc6179b90ba3f6b4662"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:DVEKBN64BKRT3D4X7VGY5MVPRL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlinear estimates for traveling wave solutions of reaction diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Li-Chang Hung, Xian Liao","submitted_at":"2019-07-12T16:20:26Z","abstract_excerpt":"In this paper we will establish nonlinear a priori lower and upper bounds for the solutions to a large class of equations which arise from the study of traveling wave solutions of reaction-diffusion equations, and we will apply our nonlinear bounds to the Lotka-Volterra system of two competing species as examples. The idea used in a series of papers \\cite{NBMP-Discrete,JDE-16,CPAA-16,DCDS-B-18,NBMP-n-species,DCDS-A-17} for the establishment of the linear N-barrier maximum principle will also be used in the proof."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05821","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":"2026-05-17T23:40:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WmkCESlc2Fz7Vn+RG6Bf5U0mJPrjyZtia9Q8hbhVkWb2vG4Umxz+je5YaoIp7f+l4XSHpzXsqPawlKaJmSPsAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:24:40.738064Z"},"content_sha256":"f8f030d24138cd2199e9316be0280ed7c1cc8ba98bffdbb103ef9de2c8f6d9c8","schema_version":"1.0","event_id":"sha256:f8f030d24138cd2199e9316be0280ed7c1cc8ba98bffdbb103ef9de2c8f6d9c8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DVEKBN64BKRT3D4X7VGY5MVPRL/bundle.json","state_url":"https://pith.science/pith/DVEKBN64BKRT3D4X7VGY5MVPRL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DVEKBN64BKRT3D4X7VGY5MVPRL/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":"2026-05-30T02:24:40Z","links":{"resolver":"https://pith.science/pith/DVEKBN64BKRT3D4X7VGY5MVPRL","bundle":"https://pith.science/pith/DVEKBN64BKRT3D4X7VGY5MVPRL/bundle.json","state":"https://pith.science/pith/DVEKBN64BKRT3D4X7VGY5MVPRL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DVEKBN64BKRT3D4X7VGY5MVPRL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:DVEKBN64BKRT3D4X7VGY5MVPRL","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":"2c18d572c6a6a1137b73da08e1e488f83b08df6f47349e5111697a60dbbb80ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-12T16:20:26Z","title_canon_sha256":"6748d4a18093d9cb7ea87666f5b86fa6ab35ac0207ed0d3d14c250c321b804f2"},"schema_version":"1.0","source":{"id":"1907.05821","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.05821","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"arxiv_version","alias_value":"1907.05821v1","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.05821","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"pith_short_12","alias_value":"DVEKBN64BKRT","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"DVEKBN64BKRT3D4X","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"DVEKBN64","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:f8f030d24138cd2199e9316be0280ed7c1cc8ba98bffdbb103ef9de2c8f6d9c8","target":"graph","created_at":"2026-05-17T23:40:44Z","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 will establish nonlinear a priori lower and upper bounds for the solutions to a large class of equations which arise from the study of traveling wave solutions of reaction-diffusion equations, and we will apply our nonlinear bounds to the Lotka-Volterra system of two competing species as examples. The idea used in a series of papers \\cite{NBMP-Discrete,JDE-16,CPAA-16,DCDS-B-18,NBMP-n-species,DCDS-A-17} for the establishment of the linear N-barrier maximum principle will also be used in the proof.","authors_text":"Li-Chang Hung, Xian Liao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-12T16:20:26Z","title":"Nonlinear estimates for traveling wave solutions of reaction diffusion equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05821","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":"sha256:7f749af90f0604c27d2c4a0166a88b31e3e8d349d36d2dc6179b90ba3f6b4662","target":"record","created_at":"2026-05-17T23:40:44Z","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":"2c18d572c6a6a1137b73da08e1e488f83b08df6f47349e5111697a60dbbb80ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-12T16:20:26Z","title_canon_sha256":"6748d4a18093d9cb7ea87666f5b86fa6ab35ac0207ed0d3d14c250c321b804f2"},"schema_version":"1.0","source":{"id":"1907.05821","kind":"arxiv","version":1}},"canonical_sha256":"1d48a0b7dc0aa33d8f97fd4d8eb2af8ac4ed567960a1b83837390671c28c22f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d48a0b7dc0aa33d8f97fd4d8eb2af8ac4ed567960a1b83837390671c28c22f5","first_computed_at":"2026-05-17T23:40:44.304701Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:44.304701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e4nvRayXYLHyR7Ha/P/bww1HdUp+nU2d51ZkuKWAf+IZZ+275JMb79rPVyqo8vhpiQ82v/Uuxg0qRhV6u4mPCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:44.305262Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.05821","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f749af90f0604c27d2c4a0166a88b31e3e8d349d36d2dc6179b90ba3f6b4662","sha256:f8f030d24138cd2199e9316be0280ed7c1cc8ba98bffdbb103ef9de2c8f6d9c8"],"state_sha256":"0f1798030b49bd780a51c3def603209ff54b96ba381c06fd009859a6c8eeaf0f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o0m8YKVPwEdWLtce7zQdEjAwxuBiGVEr0lwZyax+VTVuyS0gNcKgNBiL/gnnV1oxT0Fyr7XQ82xwPUgl/WHBBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T02:24:40.741432Z","bundle_sha256":"567d38d5cd3c397ff0e3439d9573491fd244519cc9ce8e7ef741a79c7b20f2f4"}}