{"as_of":"2026-08-18T00:18:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:0f35d81fd629a3727081cce1a48d0437e7b4f64a3f1c91a7cdfabe0baea77ac9","coverage":[{"denominator":64,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":64,"source":"paper_references, paper_reference_links","source_observed_at":"2026-06-30T11:51:17.422330Z","state":"measured"},{"denominator":64,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":64,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-17T06:30:58.91139+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2605.24666/citation-record","integrity":"/paper/2605.24666/integrity","json":"/paper/2605.24666/citation-record.json","paper":"/paper/2605.24666"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.747213Z","title":null,"venue":null,"work_id":"43e1be1b-6acc-4d5a-8b11-7274ad7bf249","year":1931},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4761a901cff3e5baca176d209d3f10bfc8f61f4905a5091d64e1cdeff40c2dcf","observation_id":"7d387e1a-e20d-4874-af88-224d5d5c8f49","resolution":{"observed_at":"2026-07-09T08:16:06.749177Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.750304Z","title":"Koopman and John von Neumann","venue":null,"work_id":"c6469e80-1fe1-4aee-9a1d-10b765013d18","year":1932},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:0dce15ceb83f03edfdfed82f395dbf35bf3f6e6782404a3602b17cdecccb29ad","observation_id":"1d630a8f-9918-4da2-9fc5-0f997e6270b6","resolution":{"observed_at":"2026-07-09T08:16:06.752170Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.740165Z","title":"Comparison of systems with complex behavior.Physica D: Nonlinear Phenomena, 197(1):101–133, 2004","venue":null,"work_id":"fcb26cbf-d344-4940-8800-3ad63d39be3c","year":2004},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:d92d233f007e1758eb4ea73352c6a9873c10f183de22ea79826d661b48ec60e7","observation_id":"5146de7f-1854-40d8-9466-25fb71ead2e8","resolution":{"observed_at":"2026-07-09T08:16:06.742336Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.731189Z","title":"Spectral properties of dynamical systems, model reduction and decompositions.Nonlinear Dynamics, 41(1):309–325, Aug 2005","venue":null,"work_id":"8aa7acde-b321-415f-8ba5-891317f22792","year":2005},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:661d559faaf3a8cad4ed175b124892553d8cdafe83d48339b4d9356a3ecb8c02","observation_id":"03fa8cb3-c85b-4221-834a-fcef1729d890","resolution":{"observed_at":"2026-07-09T08:16:06.732895Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.743445Z","title":"Applied koopmanism.Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4):047510, 12 2012","venue":null,"work_id":"d261dcf9-ae6f-40e2-b55d-068b0415f2c3","year":2012},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:e98b6900ea9a35ccca720b567ca64f19f27b1b5227d6ed7483403a743df978f8","observation_id":"ddf42c22-31f2-41de-9381-720332ad6768","resolution":{"observed_at":"2026-07-09T08:16:06.746134Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.733758Z","title":"Analysis of fluid flows via spectral properties of the Koopman operator.Annual Review of Fluid Mechanics, 45(V olume 45, 2013):357–378, 2013","venue":null,"work_id":"50ce91ec-e002-485a-a957-c8ce7f00cac6","year":2013},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:349cb462dfa88f31bef17ba01a0ae3bb8ac4c224489476711d4bc1637da61962","observation_id":"f38a4cc7-3b3a-42d8-bc13-4180dd64ae6d","resolution":{"observed_at":"2026-07-09T08:16:06.736382Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.737328Z","title":null,"venue":null,"work_id":"7eb54b88-a3e4-420f-bda2-fe5a6fb7c547","year":2022},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:5800a257718ec20357ae9c386f65d1d1cca16544cf2ac671b6d0510c35a79ff1","observation_id":"0a68a0ec-31b0-4a2e-920d-50d87435243e","resolution":{"observed_at":"2026-07-09T08:16:06.739053Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.728177Z","title":"Brunton, Marko Budiši ´c, Eurika Kaiser, and J","venue":null,"work_id":"034cb7a2-8858-47fa-85b4-5d60a4c28f0b","year":2022},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:bbe5c2af3912e1c4834e5c534973c41267de038e30ee7959f607141f3e878b5e","observation_id":"20a7c0b1-91e5-42c6-8226-488964cffb3e","resolution":{"observed_at":"2026-07-09T08:16:06.730303Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.725611Z","title":"Colbrook","venue":null,"work_id":"dcdaf0cf-332f-4a86-88e7-22229c40a1a1","year":2024},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4977c8013babfac2cef3ba36ac15cf310047d06dbed99c08e7b5059855e6b7a3","observation_id":"f5635fdc-d5b8-46e1-a2eb-c60fbf7c757e","resolution":{"observed_at":"2026-07-09T08:16:06.727246Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.587927Z","title":"Springer Basel, Basel, 2026","venue":null,"work_id":"bceca30f-b5dc-4543-9ba2-c8bd4b2f3797","year":2026},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:d58c1d272c44e1ee376a5ee302d9483bbf881066af1aaefdcc493b4c9e61c155","observation_id":"900882c3-640e-436a-92ab-cd955c40f773","resolution":{"observed_at":"2026-07-09T08:16:06.589392Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.590261Z","title":"Brunton, Joshua L","venue":null,"work_id":"33db589f-b676-4e88-9844-5ad18d8bb85d","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:a0c28a87bd00a80301a200b60d62f5abd788bdb5e2cf45c93a75c553fd0f246c","observation_id":"4ca6bcc9-7ed6-42f1-b0c6-1025e7fee24a","resolution":{"observed_at":"2026-07-09T08:16:06.591750Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.582761Z","title":"Brunton, Lise A","venue":null,"work_id":"adfb24af-3ad9-4fee-8d49-356bb4d6b8bf","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:24425241a3e143d42a56148a9bec504a63595241f90676eae83ce5a811b6fe1f","observation_id":"8237fde8-0c3d-4d9c-9e4e-d5bf6f31dd61","resolution":{"observed_at":"2026-07-09T08:16:06.584265Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.643408Z","title":"Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations.The Journal of Chemical Physics, 146(15):154104, 2017","venue":null,"work_id":"bb5d80b5-46d8-4a2f-8b4b-c7e809ccfa19","year":2017},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4fb180b2cc56e6a59e8108f9a592c7bda2398608bf384f368d20cb8ad33c887e","observation_id":"8171e577-8e75-444f-8f0b-81e4815c7be7","resolution":{"observed_at":"2026-07-09T08:16:06.645195Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.674129Z","title":"A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition.Journal of Nonlinear Science, 25(6):1307–1346, 2015","venue":null,"work_id":"172a9738-9196-445f-8b99-323d85dcf3c5","year":2015},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:8b35e4f3f15d9b91c0718dda9b3a4c6fed07b79b6c6978be63cd5cffc16854d3","observation_id":"dfebfbc7-5dbb-44b8-a685-215ae310dba9","resolution":{"observed_at":"2026-07-09T08:16:06.676016Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.585170Z","title":"On convergence of extended dynamic mode decomposition to the Koopman operator.Journal of Nonlinear Science, 28(2):687–710, 2018","venue":null,"work_id":"b2b28958-f82d-48a4-bde3-d730c567fcf0","year":2018},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:a64c7afaeb4bb78ccc4ba7130253af91a45fef1b217bf01bdf34508721ddc46a","observation_id":"9317c658-6697-4030-964e-c86c79d5bb17","resolution":{"observed_at":"2026-07-09T08:16:06.587055Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.679267Z","title":"Sharp spectral rates for Koopman operator learning.Advances in Neural Information Processing Systems, 36:32328–32339, 2023","venue":null,"work_id":"059982a8-b47b-41ba-9f45-5f5abeb0ed75","year":2023},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:59d683122a6a320d9223f8c2ee88644f829a544fa0c6788ab945e4e506f8a1e3","observation_id":"1191b576-9cd7-40d9-86ba-34fc538353f5","resolution":{"observed_at":"2026-07-09T08:16:06.681092Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.579943Z","title":"Finite-data error bounds for Koopman-based prediction and control.Journal of Nonlinear Science, 33(1):14, 2023","venue":null,"work_id":"eb58f635-1cae-4285-a2dd-608922d6cfa0","year":2023},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:33b78285842ff3e3762e2895e9f929875dc5ed98e9a2564357a3c21a7f001100","observation_id":"a95e410a-f1a4-49d7-85d5-115fd632d74f","resolution":{"observed_at":"2026-07-09T08:16:06.581920Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.612703Z","title":"Colbrook and Alex Townsend","venue":null,"work_id":"3f589ecc-bf9e-44ac-97c1-4c53a90e9032","year":2024},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:c1961afeeed94d2489d5cae39c51e71c74e4b4e0620460c6afca73cfc6228b29","observation_id":"05d0f7bd-bb2f-4bd5-9753-fef53f957a33","resolution":{"observed_at":"2026-07-09T08:16:06.614147Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.657543Z","title":"Koopman operators for estimation and control of dynamical systems.Annual Review of Control, Robotics, and Autonomous Systems, 4(1):59–87, 2021","venue":null,"work_id":"11f725d8-8c45-46e0-9f42-3ae09fbc1a34","year":2021},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:547eaff2faf9cd3eca324ad016a7dc9a5c110d6f4a761dc5ca40f56cf243dc98","observation_id":"9ff6c9c2-3e29-4719-9b23-a4d9bc6e950a","resolution":{"observed_at":"2026-07-09T08:16:06.659426Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.676805Z","title":"Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis.Advances in neural information processing systems, 29, 2016","venue":null,"work_id":"7645227c-25e9-48fa-a121-c413530bc866","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:cddef98685bd072979019b4af34df68bcc3ed8d2ed2ec1fe4ccb5a8ebd2ec972","observation_id":"7433dea7-e3c7-4908-829a-d3d0ce95878f","resolution":{"observed_at":"2026-07-09T08:16:06.678568Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.704727Z","title":"Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator.SIAM Journal on Applied Dynamical Systems, 16(4):2096–2126, 2017","venue":null,"work_id":"49ce83a2-d42a-4e60-b05b-5efeee017d89","year":2096},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:8368d440730888aee51bdd8183e6f754a9656ec0711a0338c3b91e62f302d252","observation_id":"1b512acb-0677-4413-b3c2-003d163412ba","resolution":{"observed_at":"2026-07-09T08:16:06.706249Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.668538Z","title":"Deep learning enhanced dynamic mode decomposition.Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(3):033116, 2022","venue":null,"work_id":"e875ee10-9ef7-43e1-ae63-68e8ee3eb82e","year":2022},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:5f4c72ca3cbac89ed3c454425912ccb45c9c5c495a41e78d0dd90053d7f73e22","observation_id":"b79c0ec6-5506-42d1-9ce9-057660b16fda","resolution":{"observed_at":"2026-07-09T08:16:06.670551Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.671380Z","title":"Learning Koopman invariant subspaces for dynamic mode decomposition.Advances in neural information processing systems, 30, 2017","venue":null,"work_id":"c3a2abd2-6e04-49f8-b3d6-44a344c12742","year":2017},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:c69bacb6d2ed70c5e73c8ae485cad57714a4e17b1cc4a6c04761912e0b3fe2c2","observation_id":"b1a2eb6c-9d4b-4f92-ba4f-59d635af63aa","resolution":{"observed_at":"2026-07-09T08:16:06.673278Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.597510Z","title":"Extended dynamic mode decomposi- tion with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator","venue":null,"work_id":"3865e059-6f2f-4247-946a-0561ae4ba10a","year":2017},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:0196446502b6be931c189af9d3985031c50e52972758931117677b020bb5cef2","observation_id":"66dfac56-b792-4c61-863b-d25470b8eb86","resolution":{"observed_at":"2026-07-09T08:16:06.599145Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.654845Z","title":"Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics.The Journal of chemical physics, 148(24):241703, 2018","venue":null,"work_id":"c0cb79b4-e521-4c3a-969f-8546f7258930","year":2018},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:b4275756ed0247e9c4c082835b7940dc2c55da81844c7ba408055457af3e329c","observation_id":"0cd588ec-2263-4b42-b5f3-635f8334b52a","resolution":{"observed_at":"2026-07-09T08:16:06.656598Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.692727Z","title":"Koopman operator learning using invertible neural networks","venue":null,"work_id":"68d87664-078b-487a-816d-fe347441af4b","year":2024},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:3008f12a41f3b34e6f6a479a6992e2f5a41293c5ef5d84d5862c02190dcbb402","observation_id":"c3e35f5f-00b8-4640-a43a-7217ea2931ae","resolution":{"observed_at":"2026-07-09T08:16:06.694272Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-10T21:07:42.854288Z","title":"Learning deep neural network representations for Koopman operators of nonlinear dynamical systems","venue":null,"work_id":"f5e67dd7-1a63-4a9d-bb49-25657bba5360","year":2019},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:d489c827c15cad600840650975c80c531f55791a8dd78af23281f238e38584a1","observation_id":"116c6a50-6d18-4df1-ba7c-e3c719a9b179","resolution":{"observed_at":"2026-07-09T08:16:06.685756Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.695068Z","title":"Brunton, and Matthew J","venue":null,"work_id":"efdab92d-192f-4c27-9c64-85bff635d7fd","year":2026},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:d13c5d870147e0ecfe2a434ebf21c5b986e866c4202d5aa3dc01ee7929179981","observation_id":"d12efac8-9890-4ee6-8edd-3cc3566f1aaf","resolution":{"observed_at":"2026-07-09T08:16:06.696601Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.702075Z","title":"Learning Koopman eigenfunctions and invariant subspaces from data: Symmetric subspace decomposition.IEEE Transactions on Automatic Control, 67(7):3442–3457, 2021","venue":null,"work_id":"0c1c9d78-d8eb-47f1-8409-956f84213c61","year":2021},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:053585b9c1d6d693caf851246c4376e199e8af70988d318557076ccdaa2dcbbc","observation_id":"eb23aba2-27de-4ddd-b120-7f1725df9d7b","resolution":{"observed_at":"2026-07-09T08:16:06.703943Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.717134Z","title":"Generalizing dynamic mode decomposition: Balancing accuracy and expressiveness in Koopman approximations.Automatica, 153:111001, 2023","venue":null,"work_id":"6e1cada3-ca31-4c65-932e-8e15a942cdf7","year":2023},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:e22f243978f6447c762501224d32c05dd830eda45198d146eb751883e90a726e","observation_id":"a90a3d3b-658b-4267-be3b-5c05c4bdd199","resolution":{"observed_at":"2026-07-09T08:16:06.719152Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.720039Z","title":"Courier Corporation, 2004","venue":null,"work_id":"1ffcc27c-666a-4fef-87c7-9cc69cfce692","year":2004},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:5f5ed8e22cf2ae939897ffe5609a315642fadcbb01ac5881c2f5f49398d9fcf7","observation_id":"2b929588-f0a7-4580-ac87-a13ca757f3b4","resolution":{"observed_at":"2026-07-09T08:16:06.721922Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.714346Z","title":"Finite approximation for the Frobenius-Perron operator","venue":null,"work_id":"be037290-a8e7-49f8-862e-8c5d8700d071","year":1976},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:19715b0b4f251d5b91580cf9cf19581065bd9bd0e06c9c0ce57b91b6c48af20b","observation_id":"0e269a57-afee-4bfd-8959-e01c3b5346cf","resolution":{"observed_at":"2026-07-09T08:16:06.716129Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.686673Z","title":"On the numerical approximation of the Perron-Frobenius and Koopman operator.Journal of Computational Dynamics, 3(1):51–79, 2016","venue":null,"work_id":"c35cfb75-8cf9-459f-884d-950bfe36588a","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:6a733599eebad23a3c1f8479d746861ed17b6c38f3c30dd73414a4d229004896","observation_id":"ef8043ac-c891-4222-8383-ebe97f63eb24","resolution":{"observed_at":"2026-07-09T08:16:06.689077Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.619662Z","title":"The anatomy of a large-scale hypertextual web search engine.Computer Networks and ISDN Systems, 30(1):107–117, 1998","venue":null,"work_id":"44aefe9d-0b92-4c89-a2e9-91ba75f68982","year":1998},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:1f9afd7e915799fb3acfeca1612959e33d41e86b79f62a006e0cf1bf84541908","observation_id":"12532d88-ecd5-43c1-9029-803891b7b60b","resolution":{"observed_at":"2026-07-09T08:16:06.622052Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.636171Z","title":"Deeper inside PageRank.Internet Mathematics, 1(3):335–380, 2004","venue":null,"work_id":"d63e230c-9c6e-4a3a-88a4-3dbf91f0b81c","year":2004},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:98b065b26405b1261e7c1510d70072f039341dc43e4e55eaeee3ddaf8bc4bc71","observation_id":"0fabf3b1-b40a-4196-8e8d-f69d42b0d6df","resolution":{"observed_at":"2026-07-09T08:16:06.640050Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.650138Z","title":null,"venue":null,"work_id":"93f7143e-7c12-4551-8968-656952f8bc14","year":2006},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:20d707b99fb613449a97b164f1e101afeecb14bdb0b54bbb90dc6e87cbab6894","observation_id":"d72aa33e-0566-4081-8dd2-47153eec4dd5","resolution":{"observed_at":"2026-07-09T08:16:06.651665Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.648122Z","title":null,"venue":null,"work_id":"0866a6ad-19ce-452f-b5f5-cc36323d950e","year":1989},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4f8057d5185590b67e2e5ecc77ab375501a3bfb27c4bc6de266dba3fefeb003b","observation_id":"7ffcb27f-b7b7-485a-b166-96c55d46da3a","resolution":{"observed_at":"2026-07-09T08:16:06.649482Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.577409Z","title":"Simon and Albert Ando","venue":null,"work_id":"ee753f3f-dc20-4e0b-8272-356f8f55cb20","year":1961},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:2650f92ab2b57d26074d4bd817a9f4f3a1ef2c902c54d9e4184ef2123a7d5bc1","observation_id":"cea995c7-9613-4d48-9759-68e066f6adf2","resolution":{"observed_at":"2026-07-09T08:16:06.578937Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.660150Z","title":"New York : Academic Press, New York (State), United States, 1977","venue":null,"work_id":"3d2a7d1e-2dbf-452e-80de-07015731ef2a","year":1977},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:2c4a0650d4ae1e07c61bff40bc5c96058ee25f03d0b844a5e6f93dddbc8d3f32","observation_id":"df7e432b-910c-4057-a377-d27a65b52379","resolution":{"observed_at":"2026-07-09T08:16:06.662402Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.709308Z","title":"Haveliwala","venue":null,"work_id":"8d6b9c1a-fc7e-4d5a-ae1f-e4f208ca52ef","year":2003},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:70009a167303f0204ece0982fe270e860d1814c1512690e35e0fc3fa2012d30f","observation_id":"3fd9c477-0d60-4928-a7f8-e66e7d70c77e","resolution":{"observed_at":"2026-07-09T08:16:06.710983Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.711846Z","title":"Local graph partitioning using PageRank vectors","venue":null,"work_id":"d5cdfa7a-388c-4078-9b56-6cfc1ce1a116","year":2006},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:7c9ed946fc2fa8a94ce5150529dc5e6197849fb4f822c467c241379931e54e05","observation_id":"3df93ff6-c382-4111-aebd-7c4cf814ceb3","resolution":{"observed_at":"2026-07-09T08:16:06.713530Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.665588Z","title":"A tutorial on spectral clustering.Statistics and Computing, 17(4):395–416","venue":null,"work_id":"f332d087-f23e-40b2-909c-347636981986","year":2007},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:0f1d9327a4fe4eac89fdec2d4c081f7b07343f8805bf6590bea03b9328cbe8a4","observation_id":"c14b2a61-c223-48b7-9fb6-d0086b7c70d0","resolution":{"observed_at":"2026-07-09T08:16:06.667604Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.640888Z","title":"Community detection in graphs.Physics Reports, 486(3):75–174, 2010","venue":null,"work_id":"ba3b6e36-427c-4365-84d7-f364763f863e","year":2010},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:1b0a53ae3250975c73ede1afed68737de4a459a43aa642eaefe8a4ae6164acc9","observation_id":"98407369-29c5-47ca-a6cd-caff439674ce","resolution":{"observed_at":"2026-07-09T08:16:06.642579Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.652404Z","title":"Robust Perron cluster analysis in conformation dynamics.Linear Algebra and its Applications, 398:161–184, 2005","venue":null,"work_id":"38b0fc9b-3a25-488c-bb5d-5c3453a2a537","year":2005},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:e65ac864620f27e818c6ecc584f5b3f9603d4eb0447e5e5a271a539d0388c184","observation_id":"f4511c1a-45b2-4b1c-b2fd-4d166ea50f28","resolution":{"observed_at":"2026-07-09T08:16:06.654060Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.722768Z","title":"Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification.Advances in Data Analysis and Classification, 7(2):147–179, 2013","venue":null,"work_id":"f794de71-5141-4809-b567-6b6a32a08ac2","year":2013},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:99ab28566e870ae9f50558701e1383d854027aa24c5ea2fcff75aa447fb1aef3","observation_id":"0f192cd0-cc96-41bd-874b-f2a301f1c3e5","resolution":{"observed_at":"2026-07-09T08:16:06.724791Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.592525Z","title":"Finding communities in directed networks by pager- ank random walk induced network embedding.Physica A: Statistical Mechanics and its Applications, 389(12):2443–2454, 2010","venue":null,"work_id":"2424007f-e9f1-4b7b-a148-1d5b85e88fef","year":2010},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:1d32758cec9bc5c977480d738123e21f9b6b78ed8b9049b4bcaaa4e68c68fadb","observation_id":"bf23162c-d19e-46c3-b33c-468ee902196d","resolution":{"observed_at":"2026-07-09T08:16:06.594212Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.574691Z","title":"Golub and Charles F","venue":null,"work_id":"a4f04e3b-c932-4e90-afb0-c61b3b605a69","year":2013},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4903a77520eea70e46d4ac78c87d60e98b03848ddcd505b4e5a9456c228ba469","observation_id":"1abfaa4f-8d69-4269-8224-f4c2a28befb2","resolution":{"observed_at":"2026-07-09T08:16:06.576513Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.707136Z","title":null,"venue":null,"work_id":"2ee7a475-ba17-4ec3-91af-91124be5a146","year":2001},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4b4587786a87afc4350d20cb803a4b773f6c7f8f2563fc86c0b88c09661887b3","observation_id":"aba6da05-0478-4f14-81ec-67c166b98368","resolution":{"observed_at":"2026-07-09T08:16:06.708521Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.630760Z","title":"Koopman operator, geometry, and learning of dynamical systems.Notices of the American Mathematical Society, 68(7):1087–1105, 2021","venue":null,"work_id":"e72a99ae-0fdf-4a4d-a739-f00578620749","year":2021},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4ce90fdaa67b7a5fd00654e035cb06edcfd0d114d3002bc586a0201b9975a856","observation_id":"3e2ceee9-e779-429f-8950-b60e3171d4c8","resolution":{"observed_at":"2026-07-09T08:16:06.632711Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.633486Z","title":"Ramachandran, C","venue":null,"work_id":"e89640bc-e533-4d72-a9d8-c7ab669263cf","year":1963},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:04ac04217a0068c870a9ecdf209f4ae817aaef216afe71af8058d4b46dffab4f","observation_id":"daf86968-6aa2-45a0-88f2-e282325f5e71","resolution":{"observed_at":"2026-07-09T08:16:06.635421Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.615035Z","title":"Identification of slow molecular order parameters for Markov model construction.The Journal of Chemical Physics, 139(1):015102, 2013","venue":null,"work_id":"62efafcb-d106-482f-b0ad-3f254415d01b","year":2013},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4b7e3d0df31882913dc24d3ba1b34f1c5df7270a88ab1948074a310574391a0b","observation_id":"0b73632b-fb80-4a32-a3bc-55eb09f22b1c","resolution":{"observed_at":"2026-07-09T08:16:06.618007Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.610148Z","title":"Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds, 2024","venue":null,"work_id":"949a41ac-de6f-4d07-9333-d3e9f101a2a1","year":2024},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:eec990af27870cc1743d0264cf5adb72f882444c64c58b0ba064d0e9bb8d4ed5","observation_id":"585626f0-9dfb-4131-8eda-26fe23a367a8","resolution":{"observed_at":"2026-07-09T08:16:06.611904Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.681876Z","title":"Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems.Nonlinear Dynamics, 112(3):2037– 2061, 2024","venue":null,"work_id":"e53ae917-00d4-46a3-965d-1b0f3238f52e","year":2037},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:084d849accd134de6bc8a1d21978e4ac65fd10777b26a95f6418c14a71ee41f4","observation_id":"89ab7c2c-036d-40a1-9481-4d048cb7197b","resolution":{"observed_at":"2026-07-09T08:16:06.683516Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.603873Z","title":"Dynamic mode decomposition of numerical and experimental data.Journal of Fluid Mechanics, 656:5–28","venue":null,"work_id":"f29f6655-edb2-49d9-8be9-d63101ef0a5a","year":2010},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:90abd8ccb9d25b6f28ecc23cd6f9964aa2e5c70a22648c7472ff575ffd05eb18","observation_id":"754b8362-6f8c-4593-9c32-e3cbf14e0689","resolution":{"observed_at":"2026-07-09T08:16:06.606210Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.663141Z","title":"Spectral analysis of nonlinear flows.Journal of Fluid Mechanics, 641:115–127, 2009","venue":null,"work_id":"1fd1524b-989b-49bb-b648-043861a76023","year":2009},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:ea6df4524ce3b6e4431fa14690280c369f73deb9ff0c3dcf305de0475d6f3bf1","observation_id":"eb7898b1-be7d-4d38-ab8c-dc320b6b766f","resolution":{"observed_at":"2026-07-09T08:16:06.664829Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.690222Z","title":"Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016","venue":null,"work_id":"dc3382b1-5930-4269-8b19-7494a2d7a690","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:4684b1d70cff3f07cf741f8eb7c5fe23b37848de96a122102693c4fc28307ce7","observation_id":"72b9694e-99d0-4a45-858b-ff43eb5763c4","resolution":{"observed_at":"2026-07-09T08:16:06.691879Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.624036Z","title":"A kernel-based method for data-driven Koopman spectral analysis.Journal of Computational Dynamics, 2(2):247–265, 2016","venue":null,"work_id":"a54227ca-862b-446a-abc5-725140300377","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:935428f427e2ea56abb5229f183c6415633e81f46762eb3ff568d5017f9cdaad","observation_id":"32d755f8-4cf8-4144-839b-a358b38fdfc1","resolution":{"observed_at":"2026-07-09T08:16:06.626181Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.627087Z","title":"A quantitative analysis of Koopman operator methods for system identification and predictions.Comptes Rendus","venue":null,"work_id":"30240307-5ed4-4b2f-845d-10a7e5588832","year":2023},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:031e4c14d733f47a4e7ef9256e7a54908860e236722795eaa7a7384c10d2d8be","observation_id":"5ffb8d11-0b92-4165-878f-c1cda87ba790","resolution":{"observed_at":"2026-07-09T08:16:06.629793Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.600128Z","title":null,"venue":null,"work_id":"5a533ac3-1156-41e4-bd03-6ab61434e07c","year":2015},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":59,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:fd3ec5620d1ff278706c71110e60bf9d44433e5576d9be9936f284e4d0924dc3","observation_id":"212bd898-d9ed-4ea3-9ffd-32f155d90b09","resolution":{"observed_at":"2026-07-09T08:16:06.602991Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.645902Z","title":null,"venue":null,"work_id":"33365932-644a-4a4e-9b34-4d12ad401265","year":2012},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":60,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:435401ff73b5d83f5e9561b262653f68540589373dc62bb675414ba1454d5151","observation_id":"b4d0dd7c-0eeb-4931-9314-36e28a436a7c","resolution":{"observed_at":"2026-07-09T08:16:06.647420Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.699974Z","title":"Brunton, Bingni W","venue":null,"work_id":"02868e6c-a010-441b-a464-cbe84c595ecc","year":2016},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:98a13ebe09dbc98c4485cb687c67ae4f07027c9a4b7d2c9e70482332ee60631d","observation_id":"5e7bd38b-8830-4604-9810-099a287644ae","resolution":{"observed_at":"2026-07-09T08:16:06.701259Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.595058Z","title":"A prony approximation of Koopman mode decomposition","venue":null,"work_id":"68311647-0c34-4f61-91e1-fa38b9c1e9ec","year":2015},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":62,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:74af390577986f6900500d735cf371effc2fe7e0a10d09b0305942531231ebe9","observation_id":"5a23848b-0d9d-45dc-91ef-4cdb495accb2","resolution":{"observed_at":"2026-07-09T08:16:06.596554Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.697623Z","title":"(Kψ1)(x(k)) (Kψ2)(x(k)) (Kψ3)(x(k)) #⊺ =","venue":null,"work_id":"3eba4a14-5c52-483c-a38c-51cbafbb137f","year":2017},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":63,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:7d9a883b6a09113bb59c94f7da06a5cae6ebc73e2c0569c3221091b644b02ca3","observation_id":"03598c80-d067-4260-a377-0322363c4a88","resolution":{"observed_at":"2026-07-09T08:16:06.699154Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-09T08:16:06.607357Z","title":"does not requireαnear1","venue":null,"work_id":"781b9bf2-046f-4388-845d-76f3c6a0e020","year":2000},"citing_paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank","version":3},"reference_index":64,"source":"pdf_text","source_observed_at":"2026-06-30T11:51:17.422330Z"},"links":{"citing_paper":"/paper/2605.24666"},"observation_digest":"sha256:ad43db3df80b464061591da2f8266b9cbd170dcee2ea80f015fb76646073581b","observation_id":"802db1dc-9437-4a62-8a36-77c5d8bb5582","resolution":{"observed_at":"2026-07-09T08:16:06.609200Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2605.24666","last_updated":"2026-05-27T03:46:29Z","latest_version":3,"primary_category":"math.DS","snapshot_observed_at":"2026-07-06T23:34:43.210149Z","submitted_at":"2026-05-23T17:09:02Z","title":"Finding Koopman Invariant Subspaces via Personalized PageRank"},"reference_resolution":{"displayed":64,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":7,"verified_exact":0,"verified_fuzzy":57},"total_outbound_references":64},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-17T06:30:58.91139+00:00","source":"crossref"},{"observed_at":"2026-08-17T06:30:54.323127+00:00","source":"retraction_watch"}],"thesis":"As of 18 August 2026, this Paper Citation Record lists 64 of 64 outbound references and 0 inbound Pith citation observations for arXiv:2605.24666."}