{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6RUOTYQRSF7PIDDH6OQL2FV3GZ","short_pith_number":"pith:6RUOTYQR","canonical_record":{"source":{"id":"1504.04665","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-18T00:21:01Z","cross_cats_sorted":[],"title_canon_sha256":"248a73db561a9a1747027d9f6419d9845b8ba3314b7d0e7fbf55f42b52e60489","abstract_canon_sha256":"3df0cf3df13b14056d6742f710e49b8ef642f031e7e79a14c1dbb0ac6298bbc7"},"schema_version":"1.0"},"canonical_sha256":"f468e9e211917ef40c67f3a0bd16bb3642dc7ffb60e6dd4abb7ba03c1b74cf26","source":{"kind":"arxiv","id":"1504.04665","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04665","created_at":"2026-05-18T02:18:30Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04665v1","created_at":"2026-05-18T02:18:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04665","created_at":"2026-05-18T02:18:30Z"},{"alias_kind":"pith_short_12","alias_value":"6RUOTYQRSF7P","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6RUOTYQRSF7PIDDH","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6RUOTYQR","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6RUOTYQRSF7PIDDH6OQL2FV3GZ","target":"record","payload":{"canonical_record":{"source":{"id":"1504.04665","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-18T00:21:01Z","cross_cats_sorted":[],"title_canon_sha256":"248a73db561a9a1747027d9f6419d9845b8ba3314b7d0e7fbf55f42b52e60489","abstract_canon_sha256":"3df0cf3df13b14056d6742f710e49b8ef642f031e7e79a14c1dbb0ac6298bbc7"},"schema_version":"1.0"},"canonical_sha256":"f468e9e211917ef40c67f3a0bd16bb3642dc7ffb60e6dd4abb7ba03c1b74cf26","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:30.768022Z","signature_b64":"Qthu4eV1jEWeTCVPdx/LV+KFbjl0i9eEXe2OA41CmfifwJyLsfY1HKRq+Tg82LvVAc8mf2J8KSefJfnCQhp/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f468e9e211917ef40c67f3a0bd16bb3642dc7ffb60e6dd4abb7ba03c1b74cf26","last_reissued_at":"2026-05-18T02:18:30.767576Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:30.767576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.04665","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":"2026-05-18T02:18:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RvS6m/xu3Qs1lLmfgUNNlZdOOa8POi6gI8Nw5aeNycCmM27HeuzxuTbMclBmLjidS9OQ+tov2BbxUtc2EpdUCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:14:23.372245Z"},"content_sha256":"e70ca261ee10ab991010b46235ee291931fcad53a05b93e2b353cf6cf5eb94ff","schema_version":"1.0","event_id":"sha256:e70ca261ee10ab991010b46235ee291931fcad53a05b93e2b353cf6cf5eb94ff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6RUOTYQRSF7PIDDH6OQL2FV3GZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonuniform $(h,k,\\mu,\\nu)$-Dichotomy with Applications to Nonautonomous Dynamical Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jimin Zhang, Liu Yang, Meng Fan","submitted_at":"2015-04-18T00:21:01Z","abstract_excerpt":"The paper develops and studies a very general notion of dichotomy, referred to as \"nonuniform $(h,k,\\mu,\\nu)$-dichotomy\". The new notion contains as special cases most versions of dichotomy existing in the literature. The paper then provides corresponding new versions of robustness, Hartman-Grobman theorem, and stable manifold theorem for nonautonomous dynamical systems in Banach spaces in term of the nonuniform $(h,k,\\mu,\\nu)$-dichotomy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04665","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":"2026-05-18T02:18:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a2FH4UC1kNssmSGOaQBh+lDv344YnQbwUOa4P2lRDcjLQIKbkSotzzAnYYoJ1brO4fl9kcPCToPtJ0qfLD+ACQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:14:23.372601Z"},"content_sha256":"7f45946ddfd3d7dc56d259e1eae8267b5cffb4b9b98c234a32287abab887c1f8","schema_version":"1.0","event_id":"sha256:7f45946ddfd3d7dc56d259e1eae8267b5cffb4b9b98c234a32287abab887c1f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ/bundle.json","state_url":"https://pith.science/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ/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":"2026-06-02T16:14:23Z","links":{"resolver":"https://pith.science/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ","bundle":"https://pith.science/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ/bundle.json","state":"https://pith.science/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6RUOTYQRSF7PIDDH6OQL2FV3GZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6RUOTYQRSF7PIDDH6OQL2FV3GZ","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":"3df0cf3df13b14056d6742f710e49b8ef642f031e7e79a14c1dbb0ac6298bbc7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-18T00:21:01Z","title_canon_sha256":"248a73db561a9a1747027d9f6419d9845b8ba3314b7d0e7fbf55f42b52e60489"},"schema_version":"1.0","source":{"id":"1504.04665","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04665","created_at":"2026-05-18T02:18:30Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04665v1","created_at":"2026-05-18T02:18:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04665","created_at":"2026-05-18T02:18:30Z"},{"alias_kind":"pith_short_12","alias_value":"6RUOTYQRSF7P","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6RUOTYQRSF7PIDDH","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6RUOTYQR","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:7f45946ddfd3d7dc56d259e1eae8267b5cffb4b9b98c234a32287abab887c1f8","target":"graph","created_at":"2026-05-18T02:18:30Z","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 develops and studies a very general notion of dichotomy, referred to as \"nonuniform $(h,k,\\mu,\\nu)$-dichotomy\". The new notion contains as special cases most versions of dichotomy existing in the literature. The paper then provides corresponding new versions of robustness, Hartman-Grobman theorem, and stable manifold theorem for nonautonomous dynamical systems in Banach spaces in term of the nonuniform $(h,k,\\mu,\\nu)$-dichotomy.","authors_text":"Jimin Zhang, Liu Yang, Meng Fan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-18T00:21:01Z","title":"Nonuniform $(h,k,\\mu,\\nu)$-Dichotomy with Applications to Nonautonomous Dynamical Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04665","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:e70ca261ee10ab991010b46235ee291931fcad53a05b93e2b353cf6cf5eb94ff","target":"record","created_at":"2026-05-18T02:18:30Z","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":"3df0cf3df13b14056d6742f710e49b8ef642f031e7e79a14c1dbb0ac6298bbc7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-18T00:21:01Z","title_canon_sha256":"248a73db561a9a1747027d9f6419d9845b8ba3314b7d0e7fbf55f42b52e60489"},"schema_version":"1.0","source":{"id":"1504.04665","kind":"arxiv","version":1}},"canonical_sha256":"f468e9e211917ef40c67f3a0bd16bb3642dc7ffb60e6dd4abb7ba03c1b74cf26","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f468e9e211917ef40c67f3a0bd16bb3642dc7ffb60e6dd4abb7ba03c1b74cf26","first_computed_at":"2026-05-18T02:18:30.767576Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:30.767576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qthu4eV1jEWeTCVPdx/LV+KFbjl0i9eEXe2OA41CmfifwJyLsfY1HKRq+Tg82LvVAc8mf2J8KSefJfnCQhp/Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:30.768022Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.04665","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e70ca261ee10ab991010b46235ee291931fcad53a05b93e2b353cf6cf5eb94ff","sha256:7f45946ddfd3d7dc56d259e1eae8267b5cffb4b9b98c234a32287abab887c1f8"],"state_sha256":"ff0d44b00df550ae20ac12b76599f77b3c8ac21324820e48ca6615c24af04128"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"00YZLAwZVLKhUcG9LM1783qmJDzkFqZqoG3Q0LOjbJeOfrLNSPcjrEdyrf4dKIiw3+Aj+Lxtd1ohthvC1dANBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T16:14:23.374812Z","bundle_sha256":"0d1a7d76a78f591377061bc5f326dc721eaf9c39d1d982e38ffb5bd7b3397e50"}}