{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:Y4SLLM7UY4GNNKWWGJHUUCX66Z","short_pith_number":"pith:Y4SLLM7U","canonical_record":{"source":{"id":"1406.1474","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-06-05T18:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"1bc6c95d340bfdb62893678e5701ff9843b186069ded49aa0b769e5ccf69eebc","abstract_canon_sha256":"ead2b466e1b22e24773789490ec0011e15c3feedae476a46103adc0667be3b71"},"schema_version":"1.0"},"canonical_sha256":"c724b5b3f4c70cd6aad6324f4a0afef651f9e30f52e1e0e2d5d4c39489a6bedf","source":{"kind":"arxiv","id":"1406.1474","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.1474","created_at":"2026-05-18T02:50:23Z"},{"alias_kind":"arxiv_version","alias_value":"1406.1474v1","created_at":"2026-05-18T02:50:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.1474","created_at":"2026-05-18T02:50:23Z"},{"alias_kind":"pith_short_12","alias_value":"Y4SLLM7UY4GN","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"Y4SLLM7UY4GNNKWW","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"Y4SLLM7U","created_at":"2026-05-18T12:28:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:Y4SLLM7UY4GNNKWWGJHUUCX66Z","target":"record","payload":{"canonical_record":{"source":{"id":"1406.1474","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-06-05T18:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"1bc6c95d340bfdb62893678e5701ff9843b186069ded49aa0b769e5ccf69eebc","abstract_canon_sha256":"ead2b466e1b22e24773789490ec0011e15c3feedae476a46103adc0667be3b71"},"schema_version":"1.0"},"canonical_sha256":"c724b5b3f4c70cd6aad6324f4a0afef651f9e30f52e1e0e2d5d4c39489a6bedf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:23.217419Z","signature_b64":"/x5UYmyTVQD4rbAHbiwrtQmnI9YuvCNCs8O3GyDw9m1CI/moSECAfa9bNEb4dNHIqQOGLJkkftxDNH2gPZAIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c724b5b3f4c70cd6aad6324f4a0afef651f9e30f52e1e0e2d5d4c39489a6bedf","last_reissued_at":"2026-05-18T02:50:23.216957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:23.216957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.1474","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":"2026-05-18T02:50:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c+VcjsAASG4nkrNRbCmoS80igPqFaNYYxNi6m3z040JlTY5j5wHF9BLDT5TAImhqDmcNb98wxZcQ8S0bJFK1CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T14:41:36.539666Z"},"content_sha256":"4fcf5c8e8cfccd5db75b884f48b3f60cd3bf9d8906bb9e249a4d6ee6085f2b4b","schema_version":"1.0","event_id":"sha256:4fcf5c8e8cfccd5db75b884f48b3f60cd3bf9d8906bb9e249a4d6ee6085f2b4b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:Y4SLLM7UY4GNNKWWGJHUUCX66Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Three Generalizations of Contraction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Eduardo D. Sontag, Michael Margaliot, Tamir Tuller","submitted_at":"2014-06-05T18:44:25Z","abstract_excerpt":"We introduce three forms of generalized contraction (GC). Roughly speaking, these are motivated by allowing contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asymptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior.\n  We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are GC."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1474","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":"2026-05-18T02:50:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZeeBaLCsyntkxMiGkQMNAGeOYvWHgXBDfsLM6koHFPIwCeXFUw9OsG+RyV6pQ6qpQPGh31njLX6SCIG9H240DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T14:41:36.540407Z"},"content_sha256":"be2f1e3e411fb2e8888560857591570657d6f30ae0c078e82f1d2c572a92aa9d","schema_version":"1.0","event_id":"sha256:be2f1e3e411fb2e8888560857591570657d6f30ae0c078e82f1d2c572a92aa9d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z/bundle.json","state_url":"https://pith.science/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z/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":"2026-05-31T14:41:36Z","links":{"resolver":"https://pith.science/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z","bundle":"https://pith.science/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z/bundle.json","state":"https://pith.science/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y4SLLM7UY4GNNKWWGJHUUCX66Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:Y4SLLM7UY4GNNKWWGJHUUCX66Z","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":"ead2b466e1b22e24773789490ec0011e15c3feedae476a46103adc0667be3b71","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-06-05T18:44:25Z","title_canon_sha256":"1bc6c95d340bfdb62893678e5701ff9843b186069ded49aa0b769e5ccf69eebc"},"schema_version":"1.0","source":{"id":"1406.1474","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.1474","created_at":"2026-05-18T02:50:23Z"},{"alias_kind":"arxiv_version","alias_value":"1406.1474v1","created_at":"2026-05-18T02:50:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.1474","created_at":"2026-05-18T02:50:23Z"},{"alias_kind":"pith_short_12","alias_value":"Y4SLLM7UY4GN","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"Y4SLLM7UY4GNNKWW","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"Y4SLLM7U","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:be2f1e3e411fb2e8888560857591570657d6f30ae0c078e82f1d2c572a92aa9d","target":"graph","created_at":"2026-05-18T02:50:23Z","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 introduce three forms of generalized contraction (GC). Roughly speaking, these are motivated by allowing contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asymptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior.\n  We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are GC.","authors_text":"Eduardo D. Sontag, Michael Margaliot, Tamir Tuller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-06-05T18:44:25Z","title":"On Three Generalizations of Contraction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1474","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":"sha256:4fcf5c8e8cfccd5db75b884f48b3f60cd3bf9d8906bb9e249a4d6ee6085f2b4b","target":"record","created_at":"2026-05-18T02:50:23Z","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":"ead2b466e1b22e24773789490ec0011e15c3feedae476a46103adc0667be3b71","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-06-05T18:44:25Z","title_canon_sha256":"1bc6c95d340bfdb62893678e5701ff9843b186069ded49aa0b769e5ccf69eebc"},"schema_version":"1.0","source":{"id":"1406.1474","kind":"arxiv","version":1}},"canonical_sha256":"c724b5b3f4c70cd6aad6324f4a0afef651f9e30f52e1e0e2d5d4c39489a6bedf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c724b5b3f4c70cd6aad6324f4a0afef651f9e30f52e1e0e2d5d4c39489a6bedf","first_computed_at":"2026-05-18T02:50:23.216957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:23.216957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/x5UYmyTVQD4rbAHbiwrtQmnI9YuvCNCs8O3GyDw9m1CI/moSECAfa9bNEb4dNHIqQOGLJkkftxDNH2gPZAIDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:23.217419Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.1474","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4fcf5c8e8cfccd5db75b884f48b3f60cd3bf9d8906bb9e249a4d6ee6085f2b4b","sha256:be2f1e3e411fb2e8888560857591570657d6f30ae0c078e82f1d2c572a92aa9d"],"state_sha256":"e984a7247e5b8465e328d9db3c1325e0249071545de6d47f7d9d9edb33f29cd1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"81lwowg7NvPFXStHR77SDLczuRZ4uqsfzTZ4/7pRUliUPIVdOYwx/MPxzld5qvn5SgsYlKV7I5xrvX65s6XXAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T14:41:36.544438Z","bundle_sha256":"9f60202952c1bf4db64a90cb673939b342b86b8c9affedd585cbf54a98494bd0"}}