{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:SSZDHCI7E2AIYLFZTUFTRAHJTQ","short_pith_number":"pith:SSZDHCI7","canonical_record":{"source":{"id":"1806.09880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SY","submitted_at":"2018-06-26T09:53:19Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"78c89ca0e79c45f20c1f1747e7a297ecc02c2cfdd1d66c63246f1ba3ebd1ff6c","abstract_canon_sha256":"ed0cf005e573d93fbef67a56a6ffbbf74ac62bb5949a4766035cf34cb865e411"},"schema_version":"1.0"},"canonical_sha256":"94b233891f26808c2cb99d0b3880e99c2276b8c6c53caaa53b472063eeb22bf6","source":{"kind":"arxiv","id":"1806.09880","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.09880","created_at":"2026-05-17T23:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1806.09880v1","created_at":"2026-05-17T23:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.09880","created_at":"2026-05-17T23:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"SSZDHCI7E2AI","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SSZDHCI7E2AIYLFZ","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SSZDHCI7","created_at":"2026-05-18T12:32:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:SSZDHCI7E2AIYLFZTUFTRAHJTQ","target":"record","payload":{"canonical_record":{"source":{"id":"1806.09880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SY","submitted_at":"2018-06-26T09:53:19Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"78c89ca0e79c45f20c1f1747e7a297ecc02c2cfdd1d66c63246f1ba3ebd1ff6c","abstract_canon_sha256":"ed0cf005e573d93fbef67a56a6ffbbf74ac62bb5949a4766035cf34cb865e411"},"schema_version":"1.0"},"canonical_sha256":"94b233891f26808c2cb99d0b3880e99c2276b8c6c53caaa53b472063eeb22bf6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:58.060333Z","signature_b64":"+IGVCD8WzItwd5jPyhnd8rpTfu8tc9w2OML+rzzYQXsQLB5AOWOOFy8b5T5rDpiqPl+NlpbF4LD82wSxHE2uBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94b233891f26808c2cb99d0b3880e99c2276b8c6c53caaa53b472063eeb22bf6","last_reissued_at":"2026-05-17T23:45:58.059772Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:58.059772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.09880","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":"2026-05-17T23:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ts+6ealJhJ6/QVwYeggXgELgATVNN0NK49TGVRCDA7Ij/3uiI3PswcveSBykCmGl4KhwwDL5+dUexGx5O4ArCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:49:06.855091Z"},"content_sha256":"82a0bd6dbd78cfabe6a41617b811dc60e812177997e3d7abf637e53b94040cda","schema_version":"1.0","event_id":"sha256:82a0bd6dbd78cfabe6a41617b811dc60e812177997e3d7abf637e53b94040cda"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:SSZDHCI7E2AIYLFZTUFTRAHJTQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kolmogorov $n$-widths for linear dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"cs.SY","authors_text":"Benjamin Unger, Serkan Gugercin","submitted_at":"2018-06-26T09:53:19Z","abstract_excerpt":"Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09880","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":"2026-05-17T23:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LOfNldqsOKlSCTmMbo+FZvIjdTcX6519Fg9BiNfyBHYj8IpREI49MYuDoOj55boFNsM5dQJsCSNn5qUJdkUgBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:49:06.855742Z"},"content_sha256":"81e7d50527862d88adbfaa289e27688fcc74495f2c3e7bd9522489a62085b505","schema_version":"1.0","event_id":"sha256:81e7d50527862d88adbfaa289e27688fcc74495f2c3e7bd9522489a62085b505"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ/bundle.json","state_url":"https://pith.science/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ/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":"2026-05-28T16:49:06Z","links":{"resolver":"https://pith.science/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ","bundle":"https://pith.science/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ/bundle.json","state":"https://pith.science/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SSZDHCI7E2AIYLFZTUFTRAHJTQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:SSZDHCI7E2AIYLFZTUFTRAHJTQ","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":"ed0cf005e573d93fbef67a56a6ffbbf74ac62bb5949a4766035cf34cb865e411","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SY","submitted_at":"2018-06-26T09:53:19Z","title_canon_sha256":"78c89ca0e79c45f20c1f1747e7a297ecc02c2cfdd1d66c63246f1ba3ebd1ff6c"},"schema_version":"1.0","source":{"id":"1806.09880","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.09880","created_at":"2026-05-17T23:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1806.09880v1","created_at":"2026-05-17T23:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.09880","created_at":"2026-05-17T23:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"SSZDHCI7E2AI","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SSZDHCI7E2AIYLFZ","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SSZDHCI7","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:81e7d50527862d88adbfaa289e27688fcc74495f2c3e7bd9522489a62085b505","target":"graph","created_at":"2026-05-17T23:45:58Z","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":"Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems","authors_text":"Benjamin Unger, Serkan Gugercin","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SY","submitted_at":"2018-06-26T09:53:19Z","title":"Kolmogorov $n$-widths for linear dynamical systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09880","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":"sha256:82a0bd6dbd78cfabe6a41617b811dc60e812177997e3d7abf637e53b94040cda","target":"record","created_at":"2026-05-17T23:45:58Z","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":"ed0cf005e573d93fbef67a56a6ffbbf74ac62bb5949a4766035cf34cb865e411","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SY","submitted_at":"2018-06-26T09:53:19Z","title_canon_sha256":"78c89ca0e79c45f20c1f1747e7a297ecc02c2cfdd1d66c63246f1ba3ebd1ff6c"},"schema_version":"1.0","source":{"id":"1806.09880","kind":"arxiv","version":1}},"canonical_sha256":"94b233891f26808c2cb99d0b3880e99c2276b8c6c53caaa53b472063eeb22bf6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94b233891f26808c2cb99d0b3880e99c2276b8c6c53caaa53b472063eeb22bf6","first_computed_at":"2026-05-17T23:45:58.059772Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:58.059772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+IGVCD8WzItwd5jPyhnd8rpTfu8tc9w2OML+rzzYQXsQLB5AOWOOFy8b5T5rDpiqPl+NlpbF4LD82wSxHE2uBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:58.060333Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.09880","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82a0bd6dbd78cfabe6a41617b811dc60e812177997e3d7abf637e53b94040cda","sha256:81e7d50527862d88adbfaa289e27688fcc74495f2c3e7bd9522489a62085b505"],"state_sha256":"f1d9d8707051abeae9ff56a877345d8ce4910f2e3dae3fe2303c9147fc12c532"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nEtYnjFb3y8Y2gEwWy1zdJVkrcHfbG/lLaW6mgQLofzLIuFo897n3GILaZNKvtt28JFOY3Z3G+3R3VvCRZKvAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:49:06.859351Z","bundle_sha256":"cfb1bf8d1286cb212181258191a421b8c8ed49a2ead4d1ac636ec56393db1045"}}