{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:6QRCWEYTMANCUCQUY2X23TYOP6","short_pith_number":"pith:6QRCWEYT","canonical_record":{"source":{"id":"1409.1024","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-03T10:25:05Z","cross_cats_sorted":[],"title_canon_sha256":"2c5586a828fe0954e98e0b5fe208452615087553e115e23ec61a2c07878d1cc1","abstract_canon_sha256":"f9e94774cb2e606a91a85f996bbcf7cc92c3a29a237b0b4b3f70361c95b3271a"},"schema_version":"1.0"},"canonical_sha256":"f4222b1313601a2a0a14c6afadcf0e7faeb7ae3dc489f90eaaeb44cf8b7aa1d3","source":{"kind":"arxiv","id":"1409.1024","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1024","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1024v1","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1024","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"6QRCWEYTMANC","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6QRCWEYTMANCUCQU","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6QRCWEYT","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:6QRCWEYTMANCUCQUY2X23TYOP6","target":"record","payload":{"canonical_record":{"source":{"id":"1409.1024","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-03T10:25:05Z","cross_cats_sorted":[],"title_canon_sha256":"2c5586a828fe0954e98e0b5fe208452615087553e115e23ec61a2c07878d1cc1","abstract_canon_sha256":"f9e94774cb2e606a91a85f996bbcf7cc92c3a29a237b0b4b3f70361c95b3271a"},"schema_version":"1.0"},"canonical_sha256":"f4222b1313601a2a0a14c6afadcf0e7faeb7ae3dc489f90eaaeb44cf8b7aa1d3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:44.527904Z","signature_b64":"xZH5xuRbuu6A+mHPbNMNufbnIZJsW01AoICH3mjnqi2FabpBST9dNtxVx2er0hr0r8CU16pC6gI/096yWihVBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4222b1313601a2a0a14c6afadcf0e7faeb7ae3dc489f90eaaeb44cf8b7aa1d3","last_reissued_at":"2026-05-18T02:43:44.527510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:44.527510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.1024","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":"2026-05-18T02:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3UNJzPS3sTGjZvDZqA0wMd8F4fvNcfBaltPkEl7DV7uVkv95ulVfyNaAan5wZ80AnDjZwcQOK+/r+Tk1LdijAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T17:10:04.228149Z"},"content_sha256":"d21fa6788d4063fb4ed63f60d52c1ace5c43d78d3d988a6a15ca5a69a16bbc21","schema_version":"1.0","event_id":"sha256:d21fa6788d4063fb4ed63f60d52c1ace5c43d78d3d988a6a15ca5a69a16bbc21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:6QRCWEYTMANCUCQUY2X23TYOP6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Denis D. Patterson, John A.D. Appleby","submitted_at":"2014-09-03T10:25:05Z","abstract_excerpt":"This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to their equilibrium. Sharp conditions are also established which preserve the asymptotic behaviour of the derivative of the underlying unperturbed equation. Finally, necessary and sufficient conditions are established which enable finite difference approximations to the derivative in the stochastic equation to preserve the asymptotic behaviour of the derivative of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1024","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":"2026-05-18T02:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UWsAEKmzmOCbVkyS4s7sIFrTAG9+0PDiawHvg6JmlEzPUJHHua0GynopUqL7YlM8pZDGK92SFbac7+nHNN0pCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T17:10:04.228516Z"},"content_sha256":"888d7fd910102e5780150925585e8bc6900e53cf05457c1e835e13bb48e16e4f","schema_version":"1.0","event_id":"sha256:888d7fd910102e5780150925585e8bc6900e53cf05457c1e835e13bb48e16e4f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6QRCWEYTMANCUCQUY2X23TYOP6/bundle.json","state_url":"https://pith.science/pith/6QRCWEYTMANCUCQUY2X23TYOP6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6QRCWEYTMANCUCQUY2X23TYOP6/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":"2026-06-25T17:10:04Z","links":{"resolver":"https://pith.science/pith/6QRCWEYTMANCUCQUY2X23TYOP6","bundle":"https://pith.science/pith/6QRCWEYTMANCUCQUY2X23TYOP6/bundle.json","state":"https://pith.science/pith/6QRCWEYTMANCUCQUY2X23TYOP6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6QRCWEYTMANCUCQUY2X23TYOP6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6QRCWEYTMANCUCQUY2X23TYOP6","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":"f9e94774cb2e606a91a85f996bbcf7cc92c3a29a237b0b4b3f70361c95b3271a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-03T10:25:05Z","title_canon_sha256":"2c5586a828fe0954e98e0b5fe208452615087553e115e23ec61a2c07878d1cc1"},"schema_version":"1.0","source":{"id":"1409.1024","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1024","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1024v1","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1024","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"6QRCWEYTMANC","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6QRCWEYTMANCUCQU","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6QRCWEYT","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:888d7fd910102e5780150925585e8bc6900e53cf05457c1e835e13bb48e16e4f","target":"graph","created_at":"2026-05-18T02:43:44Z","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":"This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to their equilibrium. Sharp conditions are also established which preserve the asymptotic behaviour of the derivative of the underlying unperturbed equation. Finally, necessary and sufficient conditions are established which enable finite difference approximations to the derivative in the stochastic equation to preserve the asymptotic behaviour of the derivative of","authors_text":"Denis D. Patterson, John A.D. Appleby","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-03T10:25:05Z","title":"On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1024","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:d21fa6788d4063fb4ed63f60d52c1ace5c43d78d3d988a6a15ca5a69a16bbc21","target":"record","created_at":"2026-05-18T02:43:44Z","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":"f9e94774cb2e606a91a85f996bbcf7cc92c3a29a237b0b4b3f70361c95b3271a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-03T10:25:05Z","title_canon_sha256":"2c5586a828fe0954e98e0b5fe208452615087553e115e23ec61a2c07878d1cc1"},"schema_version":"1.0","source":{"id":"1409.1024","kind":"arxiv","version":1}},"canonical_sha256":"f4222b1313601a2a0a14c6afadcf0e7faeb7ae3dc489f90eaaeb44cf8b7aa1d3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4222b1313601a2a0a14c6afadcf0e7faeb7ae3dc489f90eaaeb44cf8b7aa1d3","first_computed_at":"2026-05-18T02:43:44.527510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:44.527510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xZH5xuRbuu6A+mHPbNMNufbnIZJsW01AoICH3mjnqi2FabpBST9dNtxVx2er0hr0r8CU16pC6gI/096yWihVBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:44.527904Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.1024","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d21fa6788d4063fb4ed63f60d52c1ace5c43d78d3d988a6a15ca5a69a16bbc21","sha256:888d7fd910102e5780150925585e8bc6900e53cf05457c1e835e13bb48e16e4f"],"state_sha256":"248caf087ee662a3e7019099b660211783e4607a0e6d25d4e47d17c798b0e75c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HPVvCfXe4YKMJdS+1+q3FUjkoWwJM6t1HZM4cG0FX76JdH+5QtezITsu1o1WGJnt7wyCN4XhwJ5ur041IEJpDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T17:10:04.230734Z","bundle_sha256":"1586f0ecd6172d70dc7d7293b0de28c94179706500ca43050a3b53abb9d99a35"}}