{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:IJLOM3UJOPMZPAEYM5VEI4JZUI","short_pith_number":"pith:IJLOM3UJ","canonical_record":{"source":{"id":"1406.5378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-06-20T13:23:39Z","cross_cats_sorted":[],"title_canon_sha256":"cb040b820465c6209b55ccb50fcc486f162578666bab248de23c25441d8b46eb","abstract_canon_sha256":"2a574478a4f7810d9173478ebd9d20b3b861e3d8e72cdaa7172cc7baf0ab9a5b"},"schema_version":"1.0"},"canonical_sha256":"4256e66e8973d9978098676a447139a2120b0b4c6b4d5cf5a4963788334c2b74","source":{"kind":"arxiv","id":"1406.5378","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5378","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5378v2","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5378","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"pith_short_12","alias_value":"IJLOM3UJOPMZ","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IJLOM3UJOPMZPAEY","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IJLOM3UJ","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:IJLOM3UJOPMZPAEYM5VEI4JZUI","target":"record","payload":{"canonical_record":{"source":{"id":"1406.5378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-06-20T13:23:39Z","cross_cats_sorted":[],"title_canon_sha256":"cb040b820465c6209b55ccb50fcc486f162578666bab248de23c25441d8b46eb","abstract_canon_sha256":"2a574478a4f7810d9173478ebd9d20b3b861e3d8e72cdaa7172cc7baf0ab9a5b"},"schema_version":"1.0"},"canonical_sha256":"4256e66e8973d9978098676a447139a2120b0b4c6b4d5cf5a4963788334c2b74","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:20.732430Z","signature_b64":"b5B+CLEJR6JqOd80U6qESbK73PADVqxv1Hvi3oUuKubxgQgmCfQtw7DAovSkV2EqN1cVH4nk0FPnlFgfOEBXAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4256e66e8973d9978098676a447139a2120b0b4c6b4d5cf5a4963788334c2b74","last_reissued_at":"2026-05-18T01:36:20.731962Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:20.731962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.5378","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nBDO7esp3CoTbvw9X2s4n70orKLlAHT+64fpD2g/UZ1ocokrKqsB/UZ2Fyg7IyaENOJkA4NO3HGXQJ+yKKFBBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:21:49.630826Z"},"content_sha256":"1fffc01eda5c93323196019a19e4691e498e956afe758ad68bc8f5961772a445","schema_version":"1.0","event_id":"sha256:1fffc01eda5c93323196019a19e4691e498e956afe758ad68bc8f5961772a445"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:IJLOM3UJOPMZPAEYM5VEI4JZUI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Kurusch Ebrahimi-Fard, Luis A. Duffaut Espinosa, W. Steven Gray","submitted_at":"2014-06-20T13:23:39Z","abstract_excerpt":"Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic set up is introduced using a new type of grading that has recent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5378","kind":"arxiv","version":2},"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":"2026-05-18T01:36:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7gYsJ8b7fg0SXxs0JhupIOdW5QrfftvEumwWPasyAirTvMhn8bQXGBxd+mCuxG14NU64SQnbdvYwAR4MFDE8CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:21:49.631400Z"},"content_sha256":"a08a156f41fb820ca38738416038e4125252543d13e01320b733aa8084ac4b2c","schema_version":"1.0","event_id":"sha256:a08a156f41fb820ca38738416038e4125252543d13e01320b733aa8084ac4b2c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI/bundle.json","state_url":"https://pith.science/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI/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":"2026-05-31T05:21:49Z","links":{"resolver":"https://pith.science/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI","bundle":"https://pith.science/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI/bundle.json","state":"https://pith.science/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJLOM3UJOPMZPAEYM5VEI4JZUI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:IJLOM3UJOPMZPAEYM5VEI4JZUI","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":"2a574478a4f7810d9173478ebd9d20b3b861e3d8e72cdaa7172cc7baf0ab9a5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-06-20T13:23:39Z","title_canon_sha256":"cb040b820465c6209b55ccb50fcc486f162578666bab248de23c25441d8b46eb"},"schema_version":"1.0","source":{"id":"1406.5378","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5378","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5378v2","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5378","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"pith_short_12","alias_value":"IJLOM3UJOPMZ","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IJLOM3UJOPMZPAEY","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IJLOM3UJ","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:a08a156f41fb820ca38738416038e4125252543d13e01320b733aa8084ac4b2c","target":"graph","created_at":"2026-05-18T01:36:20Z","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":"Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic set up is introduced using a new type of grading that has recent","authors_text":"Kurusch Ebrahimi-Fard, Luis A. Duffaut Espinosa, W. Steven Gray","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-06-20T13:23:39Z","title":"Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5378","kind":"arxiv","version":2},"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":"sha256:1fffc01eda5c93323196019a19e4691e498e956afe758ad68bc8f5961772a445","target":"record","created_at":"2026-05-18T01:36:20Z","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":"2a574478a4f7810d9173478ebd9d20b3b861e3d8e72cdaa7172cc7baf0ab9a5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-06-20T13:23:39Z","title_canon_sha256":"cb040b820465c6209b55ccb50fcc486f162578666bab248de23c25441d8b46eb"},"schema_version":"1.0","source":{"id":"1406.5378","kind":"arxiv","version":2}},"canonical_sha256":"4256e66e8973d9978098676a447139a2120b0b4c6b4d5cf5a4963788334c2b74","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4256e66e8973d9978098676a447139a2120b0b4c6b4d5cf5a4963788334c2b74","first_computed_at":"2026-05-18T01:36:20.731962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:20.731962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b5B+CLEJR6JqOd80U6qESbK73PADVqxv1Hvi3oUuKubxgQgmCfQtw7DAovSkV2EqN1cVH4nk0FPnlFgfOEBXAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:20.732430Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.5378","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1fffc01eda5c93323196019a19e4691e498e956afe758ad68bc8f5961772a445","sha256:a08a156f41fb820ca38738416038e4125252543d13e01320b733aa8084ac4b2c"],"state_sha256":"e16b1f911fc5f360c3d57a2e0aa68f8d49f5b0bea4f5502211e283ec15bae7d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QL2TVJbyqH+hQ0BTXEA5LlEuxIEc2DKHTQRgr2o8sQGoPIQ7so4zExZoTmmXos92L30zF9PF53A0pgGgoZLNAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T05:21:49.634913Z","bundle_sha256":"c500b69da7f3a425b28c80218162da5d4b188268703effebfd44936ff01a741c"}}