{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:4VNKH5B5E6ODNEKNTWJKDBF6QP","short_pith_number":"pith:4VNKH5B5","canonical_record":{"source":{"id":"1809.05289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T07:52:37Z","cross_cats_sorted":[],"title_canon_sha256":"2519296a6273c7cc4fa461880f57794e32f37246e7d00d5631b58d310c81c821","abstract_canon_sha256":"7d8e87b2417a891b0a0d905dd25cd6bf467bcb3db1b80df25947f4ecf341d2f4"},"schema_version":"1.0"},"canonical_sha256":"e55aa3f43d279c36914d9d92a184be83d97bcc022454da63484d2ed485a2bd89","source":{"kind":"arxiv","id":"1809.05289","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.05289","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"arxiv_version","alias_value":"1809.05289v1","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.05289","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"pith_short_12","alias_value":"4VNKH5B5E6OD","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"4VNKH5B5E6ODNEKN","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"4VNKH5B5","created_at":"2026-05-18T12:32:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:4VNKH5B5E6ODNEKNTWJKDBF6QP","target":"record","payload":{"canonical_record":{"source":{"id":"1809.05289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T07:52:37Z","cross_cats_sorted":[],"title_canon_sha256":"2519296a6273c7cc4fa461880f57794e32f37246e7d00d5631b58d310c81c821","abstract_canon_sha256":"7d8e87b2417a891b0a0d905dd25cd6bf467bcb3db1b80df25947f4ecf341d2f4"},"schema_version":"1.0"},"canonical_sha256":"e55aa3f43d279c36914d9d92a184be83d97bcc022454da63484d2ed485a2bd89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:44.095101Z","signature_b64":"xcOxuv9tt/M2gkcNgLCp6hqCZOffgZyjVsP9Nc6iy8o99GVp/Rl5D98OdHTEcMYRYkxmobJA3DOHOQczvT5QAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e55aa3f43d279c36914d9d92a184be83d97bcc022454da63484d2ed485a2bd89","last_reissued_at":"2026-05-18T00:05:44.094540Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:44.094540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.05289","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":"2026-05-18T00:05:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wwb8nqsiw9JRTE+hociHcLnNppmWYGmEiy0Zz1xE+wHNIX2h7v9t5dEK9dNtmJcxf6P15qRIc0wAgHLa2cDkAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:28:22.026091Z"},"content_sha256":"8250916f76b7bd70610dbef058e82409b00879fc372b04d6c13f430ffd67af15","schema_version":"1.0","event_id":"sha256:8250916f76b7bd70610dbef058e82409b00879fc372b04d6c13f430ffd67af15"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:4VNKH5B5E6ODNEKNTWJKDBF6QP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lyapunov Theory for Discrete Time Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Luca Schenato, Nicoletta Bof, Ruggero Carli","submitted_at":"2018-09-14T07:52:37Z","abstract_excerpt":"In this work, we present the equivalent of many theorems available for continuous time systems. In particular, the theory is applied to Averaging Theory and Separation of time scales. In particular the proofs developed for Averaging Theory and Separation of time scales departs from those typically used in continuous time systems that are based on twice differentiable change of variables and the multiple use of the Implicit Function Theorem and Mean Value Theorem. More specifically, by constructing a suitable Lyapunov function only Lipschitz conditions are necessary. Finally, it is shown that u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05289","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":"2026-05-18T00:05:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wDttUx4CfvDCqDBuJEp7ew7elfkixMNObIP8meg4Bj4mVwXDs/IZI60qzPb1u+UtH8KlhNBZS0QaXbUPR672Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:28:22.026482Z"},"content_sha256":"be19909a0f2c98f4210ed97ead25aa4d92e90f8c1a7f2dc4bc121725cadfd2f9","schema_version":"1.0","event_id":"sha256:be19909a0f2c98f4210ed97ead25aa4d92e90f8c1a7f2dc4bc121725cadfd2f9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP/bundle.json","state_url":"https://pith.science/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP/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":"2026-05-30T15:28:22Z","links":{"resolver":"https://pith.science/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP","bundle":"https://pith.science/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP/bundle.json","state":"https://pith.science/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4VNKH5B5E6ODNEKNTWJKDBF6QP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:4VNKH5B5E6ODNEKNTWJKDBF6QP","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":"7d8e87b2417a891b0a0d905dd25cd6bf467bcb3db1b80df25947f4ecf341d2f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T07:52:37Z","title_canon_sha256":"2519296a6273c7cc4fa461880f57794e32f37246e7d00d5631b58d310c81c821"},"schema_version":"1.0","source":{"id":"1809.05289","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.05289","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"arxiv_version","alias_value":"1809.05289v1","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.05289","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"pith_short_12","alias_value":"4VNKH5B5E6OD","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"4VNKH5B5E6ODNEKN","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"4VNKH5B5","created_at":"2026-05-18T12:32:05Z"}],"graph_snapshots":[{"event_id":"sha256:be19909a0f2c98f4210ed97ead25aa4d92e90f8c1a7f2dc4bc121725cadfd2f9","target":"graph","created_at":"2026-05-18T00:05:44Z","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 work, we present the equivalent of many theorems available for continuous time systems. In particular, the theory is applied to Averaging Theory and Separation of time scales. In particular the proofs developed for Averaging Theory and Separation of time scales departs from those typically used in continuous time systems that are based on twice differentiable change of variables and the multiple use of the Implicit Function Theorem and Mean Value Theorem. More specifically, by constructing a suitable Lyapunov function only Lipschitz conditions are necessary. Finally, it is shown that u","authors_text":"Luca Schenato, Nicoletta Bof, Ruggero Carli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T07:52:37Z","title":"Lyapunov Theory for Discrete Time Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05289","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":"sha256:8250916f76b7bd70610dbef058e82409b00879fc372b04d6c13f430ffd67af15","target":"record","created_at":"2026-05-18T00:05:44Z","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":"7d8e87b2417a891b0a0d905dd25cd6bf467bcb3db1b80df25947f4ecf341d2f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T07:52:37Z","title_canon_sha256":"2519296a6273c7cc4fa461880f57794e32f37246e7d00d5631b58d310c81c821"},"schema_version":"1.0","source":{"id":"1809.05289","kind":"arxiv","version":1}},"canonical_sha256":"e55aa3f43d279c36914d9d92a184be83d97bcc022454da63484d2ed485a2bd89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e55aa3f43d279c36914d9d92a184be83d97bcc022454da63484d2ed485a2bd89","first_computed_at":"2026-05-18T00:05:44.094540Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:44.094540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xcOxuv9tt/M2gkcNgLCp6hqCZOffgZyjVsP9Nc6iy8o99GVp/Rl5D98OdHTEcMYRYkxmobJA3DOHOQczvT5QAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:44.095101Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.05289","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8250916f76b7bd70610dbef058e82409b00879fc372b04d6c13f430ffd67af15","sha256:be19909a0f2c98f4210ed97ead25aa4d92e90f8c1a7f2dc4bc121725cadfd2f9"],"state_sha256":"ba91732874495f589b9fa2f5cb7dd5b34be4190a19bedf5b82a56c52610f87e6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gDOdtJpdnAF16w+8eqwP722aM96UFofHJyizwdR1ErAAbkY+hH73mEKMlC9Ss0rkeVPRJhdm1dMxzlC6Dg4EBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T15:28:22.029268Z","bundle_sha256":"94395c93db09de25c596fa1374f18ef8272482b2d242103f1cdf8f8136a8b87a"}}