{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:EWMJSPGG7LGHIM3GP27CZB2OQG","short_pith_number":"pith:EWMJSPGG","canonical_record":{"source":{"id":"1405.2946","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-05-12T20:25:57Z","cross_cats_sorted":[],"title_canon_sha256":"b16e32a92454525e1d1a13610609fb5293fb75d9840e00074bc8eae0875b7642","abstract_canon_sha256":"f1efe79be833c102528a27b825dcdac1bae30735ab08dc3378f5acea8d441fec"},"schema_version":"1.0"},"canonical_sha256":"2598993cc6facc7433667ebe2c874e8193944a09fef37fc4ddc34df24ce16912","source":{"kind":"arxiv","id":"1405.2946","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.2946","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"1405.2946v3","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.2946","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"EWMJSPGG7LGH","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"EWMJSPGG7LGHIM3G","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"EWMJSPGG","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:EWMJSPGG7LGHIM3GP27CZB2OQG","target":"record","payload":{"canonical_record":{"source":{"id":"1405.2946","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-05-12T20:25:57Z","cross_cats_sorted":[],"title_canon_sha256":"b16e32a92454525e1d1a13610609fb5293fb75d9840e00074bc8eae0875b7642","abstract_canon_sha256":"f1efe79be833c102528a27b825dcdac1bae30735ab08dc3378f5acea8d441fec"},"schema_version":"1.0"},"canonical_sha256":"2598993cc6facc7433667ebe2c874e8193944a09fef37fc4ddc34df24ce16912","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:40.235739Z","signature_b64":"xJxAIMy+K8uwNm3dtnDdiTDEaEfXCPNjzSkTCG/xI241f6Q7MyKaEzm1B+u7kbiFhEjziphEtbpRH6tl9hNeBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2598993cc6facc7433667ebe2c874e8193944a09fef37fc4ddc34df24ce16912","last_reissued_at":"2026-05-18T00:38:40.235137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:40.235137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.2946","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/zRj99kZSQAdzyZVm4GACDUwcX7dkfpvjaYaWAlP3WoP6thtOKBc4P3vq2aqorNyHWhBqFoyLuh/LR9Hm5TVBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T07:19:12.185793Z"},"content_sha256":"e437715bab6108034f76648a630d5a0273662c0b3cccc821126e96e07229b4c2","schema_version":"1.0","event_id":"sha256:e437715bab6108034f76648a630d5a0273662c0b3cccc821126e96e07229b4c2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:EWMJSPGG7LGHIM3GP27CZB2OQG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Integral conditions for nonuniform $\\mu$-dichotomy on the half-line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ant\\'onio J. G. Bento, C\\'esar M. Silva, Mihail Megan, Nicolae Lupa","submitted_at":"2014-05-12T20:25:57Z","abstract_excerpt":"We give necessary integral conditions and sufficient ones for the existence of a general concept of $\\mu$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $\\mu$-dichotomies using these Lyapunov functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2946","kind":"arxiv","version":3},"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":"2026-05-18T00:38:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U+FCJkDhD+/lkrCr+QHXO4T61FpT6X0AlTfzo4mso30z9Yy48UGP9bwPmi8KeaOlPC45O+bXhB+A8unBzeXBAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T07:19:12.186143Z"},"content_sha256":"7056d4fa5db49537b6bf6c17d40e1ca6af7d272cdcc911cf9231f4dc8f166ed7","schema_version":"1.0","event_id":"sha256:7056d4fa5db49537b6bf6c17d40e1ca6af7d272cdcc911cf9231f4dc8f166ed7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EWMJSPGG7LGHIM3GP27CZB2OQG/bundle.json","state_url":"https://pith.science/pith/EWMJSPGG7LGHIM3GP27CZB2OQG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EWMJSPGG7LGHIM3GP27CZB2OQG/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":"2026-06-04T07:19:12Z","links":{"resolver":"https://pith.science/pith/EWMJSPGG7LGHIM3GP27CZB2OQG","bundle":"https://pith.science/pith/EWMJSPGG7LGHIM3GP27CZB2OQG/bundle.json","state":"https://pith.science/pith/EWMJSPGG7LGHIM3GP27CZB2OQG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EWMJSPGG7LGHIM3GP27CZB2OQG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:EWMJSPGG7LGHIM3GP27CZB2OQG","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":"f1efe79be833c102528a27b825dcdac1bae30735ab08dc3378f5acea8d441fec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-05-12T20:25:57Z","title_canon_sha256":"b16e32a92454525e1d1a13610609fb5293fb75d9840e00074bc8eae0875b7642"},"schema_version":"1.0","source":{"id":"1405.2946","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.2946","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"1405.2946v3","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.2946","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"EWMJSPGG7LGH","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"EWMJSPGG7LGHIM3G","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"EWMJSPGG","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:7056d4fa5db49537b6bf6c17d40e1ca6af7d272cdcc911cf9231f4dc8f166ed7","target":"graph","created_at":"2026-05-18T00:38:40Z","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":"We give necessary integral conditions and sufficient ones for the existence of a general concept of $\\mu$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $\\mu$-dichotomies using these Lyapunov functions.","authors_text":"Ant\\'onio J. G. Bento, C\\'esar M. Silva, Mihail Megan, Nicolae Lupa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-05-12T20:25:57Z","title":"Integral conditions for nonuniform $\\mu$-dichotomy on the half-line"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2946","kind":"arxiv","version":3},"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:e437715bab6108034f76648a630d5a0273662c0b3cccc821126e96e07229b4c2","target":"record","created_at":"2026-05-18T00:38:40Z","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":"f1efe79be833c102528a27b825dcdac1bae30735ab08dc3378f5acea8d441fec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-05-12T20:25:57Z","title_canon_sha256":"b16e32a92454525e1d1a13610609fb5293fb75d9840e00074bc8eae0875b7642"},"schema_version":"1.0","source":{"id":"1405.2946","kind":"arxiv","version":3}},"canonical_sha256":"2598993cc6facc7433667ebe2c874e8193944a09fef37fc4ddc34df24ce16912","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2598993cc6facc7433667ebe2c874e8193944a09fef37fc4ddc34df24ce16912","first_computed_at":"2026-05-18T00:38:40.235137Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:40.235137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xJxAIMy+K8uwNm3dtnDdiTDEaEfXCPNjzSkTCG/xI241f6Q7MyKaEzm1B+u7kbiFhEjziphEtbpRH6tl9hNeBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:40.235739Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.2946","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e437715bab6108034f76648a630d5a0273662c0b3cccc821126e96e07229b4c2","sha256:7056d4fa5db49537b6bf6c17d40e1ca6af7d272cdcc911cf9231f4dc8f166ed7"],"state_sha256":"d4dbe6fafe5090f5d80bc466979b30bcf54f9114e984e7f3901becaa5b27c7f7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sSTelmA9ZqN+yIdOV5noxET/UyXAcyNuANyrPn9J2lJfer79nYbN8TV8qQTipcAjeplK5PBRJ4ORa5ux3i7eAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T07:19:12.188167Z","bundle_sha256":"bb968fb548195af86accf2285aa03d0daa461f418b8e6923b9f1c7ecf4d34903"}}