{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:QE22EZOPV6VPBGS2EPVK54KRVJ","short_pith_number":"pith:QE22EZOP","canonical_record":{"source":{"id":"1402.0753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-04T15:07:40Z","cross_cats_sorted":["math.AP","math.OC"],"title_canon_sha256":"d8a1795b86c8e4da8f36ccbd75f5e731da35315818d3155417e7d50c2cba42f9","abstract_canon_sha256":"10b25182bf3e9e661d95d9879398684f4cdd19c022e29ecf57feddd055965c0f"},"schema_version":"1.0"},"canonical_sha256":"8135a265cfafaaf09a5a23eaaef151aa70068bdf9e4fb5af6f6c864784097dfd","source":{"kind":"arxiv","id":"1402.0753","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.0753","created_at":"2026-05-18T03:00:12Z"},{"alias_kind":"arxiv_version","alias_value":"1402.0753v1","created_at":"2026-05-18T03:00:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.0753","created_at":"2026-05-18T03:00:12Z"},{"alias_kind":"pith_short_12","alias_value":"QE22EZOPV6VP","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"QE22EZOPV6VPBGS2","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"QE22EZOP","created_at":"2026-05-18T12:28:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:QE22EZOPV6VPBGS2EPVK54KRVJ","target":"record","payload":{"canonical_record":{"source":{"id":"1402.0753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-04T15:07:40Z","cross_cats_sorted":["math.AP","math.OC"],"title_canon_sha256":"d8a1795b86c8e4da8f36ccbd75f5e731da35315818d3155417e7d50c2cba42f9","abstract_canon_sha256":"10b25182bf3e9e661d95d9879398684f4cdd19c022e29ecf57feddd055965c0f"},"schema_version":"1.0"},"canonical_sha256":"8135a265cfafaaf09a5a23eaaef151aa70068bdf9e4fb5af6f6c864784097dfd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:12.070015Z","signature_b64":"6ewTmkTcsnPBrWHu8MBe2FUnfSYrRw261gi0VUo1HtmqducwG3ISNnj6lRsfmKS7XB5/p+W61mWsAYBFz14IAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8135a265cfafaaf09a5a23eaaef151aa70068bdf9e4fb5af6f6c864784097dfd","last_reissued_at":"2026-05-18T03:00:12.069330Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:12.069330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.0753","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":"2026-05-18T03:00:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oc7G1/kXvgPu4zZ5QUhW2AvkotNyt3n7V11PltbgKyMfz8u5R+yzrXHyXhBWUQWpW5OOV+tQ5+ihFC9a42kUBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T02:32:45.422811Z"},"content_sha256":"b650356bc228f5d8974d0a91af7119d8316ba96b5269ff0f89dc7fd6db3e4681","schema_version":"1.0","event_id":"sha256:b650356bc228f5d8974d0a91af7119d8316ba96b5269ff0f89dc7fd6db3e4681"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:QE22EZOPV6VPBGS2EPVK54KRVJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Stability of Stochastic Parametrically Forced Equations with Rank One Forcing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.OC"],"primary_cat":"math.DS","authors_text":"J.R. Torczynski, L.A. Romero, Timothy Blass","submitted_at":"2014-02-04T15:07:40Z","abstract_excerpt":"We derive simplified formulas for analyzing the stability of stochastic parametrically forced linear systems. This extends the results in [T. Blass and L.A. Romero, SIAM J. Control Optim. 51(2):1099--1127, 2013] where, assuming the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0753","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":"2026-05-18T03:00:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9T6bWSUUdfkwezFY5FmZXoJ8FOKva6rF1TqQrTl3RixfqyXRdOB3S1aG4oZJvbgPC7i/6nlWkpzqXFvln4JEDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T02:32:45.423602Z"},"content_sha256":"1b831997be710cb35994b687eef49c9efa86e381ac762f7233c51bc826136acc","schema_version":"1.0","event_id":"sha256:1b831997be710cb35994b687eef49c9efa86e381ac762f7233c51bc826136acc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QE22EZOPV6VPBGS2EPVK54KRVJ/bundle.json","state_url":"https://pith.science/pith/QE22EZOPV6VPBGS2EPVK54KRVJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QE22EZOPV6VPBGS2EPVK54KRVJ/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":"2026-05-31T02:32:45Z","links":{"resolver":"https://pith.science/pith/QE22EZOPV6VPBGS2EPVK54KRVJ","bundle":"https://pith.science/pith/QE22EZOPV6VPBGS2EPVK54KRVJ/bundle.json","state":"https://pith.science/pith/QE22EZOPV6VPBGS2EPVK54KRVJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QE22EZOPV6VPBGS2EPVK54KRVJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:QE22EZOPV6VPBGS2EPVK54KRVJ","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":"10b25182bf3e9e661d95d9879398684f4cdd19c022e29ecf57feddd055965c0f","cross_cats_sorted":["math.AP","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-04T15:07:40Z","title_canon_sha256":"d8a1795b86c8e4da8f36ccbd75f5e731da35315818d3155417e7d50c2cba42f9"},"schema_version":"1.0","source":{"id":"1402.0753","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.0753","created_at":"2026-05-18T03:00:12Z"},{"alias_kind":"arxiv_version","alias_value":"1402.0753v1","created_at":"2026-05-18T03:00:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.0753","created_at":"2026-05-18T03:00:12Z"},{"alias_kind":"pith_short_12","alias_value":"QE22EZOPV6VP","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"QE22EZOPV6VPBGS2","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"QE22EZOP","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:1b831997be710cb35994b687eef49c9efa86e381ac762f7233c51bc826136acc","target":"graph","created_at":"2026-05-18T03:00:12Z","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 derive simplified formulas for analyzing the stability of stochastic parametrically forced linear systems. This extends the results in [T. Blass and L.A. Romero, SIAM J. Control Optim. 51(2):1099--1127, 2013] where, assuming the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this ","authors_text":"J.R. Torczynski, L.A. Romero, Timothy Blass","cross_cats":["math.AP","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-04T15:07:40Z","title":"On the Stability of Stochastic Parametrically Forced Equations with Rank One Forcing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0753","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:b650356bc228f5d8974d0a91af7119d8316ba96b5269ff0f89dc7fd6db3e4681","target":"record","created_at":"2026-05-18T03:00:12Z","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":"10b25182bf3e9e661d95d9879398684f4cdd19c022e29ecf57feddd055965c0f","cross_cats_sorted":["math.AP","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-04T15:07:40Z","title_canon_sha256":"d8a1795b86c8e4da8f36ccbd75f5e731da35315818d3155417e7d50c2cba42f9"},"schema_version":"1.0","source":{"id":"1402.0753","kind":"arxiv","version":1}},"canonical_sha256":"8135a265cfafaaf09a5a23eaaef151aa70068bdf9e4fb5af6f6c864784097dfd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8135a265cfafaaf09a5a23eaaef151aa70068bdf9e4fb5af6f6c864784097dfd","first_computed_at":"2026-05-18T03:00:12.069330Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:00:12.069330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6ewTmkTcsnPBrWHu8MBe2FUnfSYrRw261gi0VUo1HtmqducwG3ISNnj6lRsfmKS7XB5/p+W61mWsAYBFz14IAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:00:12.070015Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.0753","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b650356bc228f5d8974d0a91af7119d8316ba96b5269ff0f89dc7fd6db3e4681","sha256:1b831997be710cb35994b687eef49c9efa86e381ac762f7233c51bc826136acc"],"state_sha256":"abf2c3df83250c4616ed7380a5d946b85e76ab96dd910146e56de972511dcaea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8+2Fq5lg83Q4PNWaiJvfRwKhCDROdur2VoyMl8oWRfepGa0L2JpLI2mxNLUiwqZColj2P1p2tlY15RwamJiJCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T02:32:45.427757Z","bundle_sha256":"60cc6558b5696650ada4149a94052246fb5a040634b3bba00c2bdf200654d1b2"}}