{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1998:WG544QPMOWFNRVM2JIW6GGXH7E","short_pith_number":"pith:WG544QPM","schema_version":"1.0","canonical_sha256":"b1bbce41ec758ad8d59a4a2de31ae7f9240292e18f47ef130723d6530c99b979","source":{"kind":"arxiv","id":"chao-dyn/9802006","version":1},"attestation_state":"computed","paper":{"title":"Extensive Properties of the Complex Ginzburg-Landau Equation","license":"","headline":"","cross_cats":["nlin.CD","nlin.PS","patt-sol"],"primary_cat":"chao-dyn","authors_text":"Jean-Pierre Eckmann, Pierre Collet","submitted_at":"1998-02-06T07:25:42Z","abstract_excerpt":"We study the set of solutions of the complex Ginzburg-Landau equation in $\\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(\\epsilon)$ of balls of radius $\\epsilon$ in $\\Linfty(Q_L)$. We show that the Kolmogorov $\\epsilon$-entropy per unit length, $H_\\epsilon =\\lim_{L\\to\\infty} L^{-d} \\log N_{Q_L}(\\epsilon)$ exists. In particular, we bound $H_\\epsilon$ by $\\OO(\\log(1/\\epsilon)$, which shows that the attracting set is smaller than the set o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"chao-dyn/9802006","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"chao-dyn","submitted_at":"1998-02-06T07:25:42Z","cross_cats_sorted":["nlin.CD","nlin.PS","patt-sol"],"title_canon_sha256":"ce7b43a98b43171d221a28533814e67ad869a9d5a38c805066d777e5c24130dd","abstract_canon_sha256":"cb772e9618d50738b22232a8ea7fb1c3ae7b75404ca77baf8b54dbb447b24183"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:24.591253Z","signature_b64":"yukwcBd8u7yEz7yvXDwr360OCeLFmrQZa5oJpQpVquykHeLmhcuF/dg2RUb3vaLyMqxOaHEQHof3NsM25l6hCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1bbce41ec758ad8d59a4a2de31ae7f9240292e18f47ef130723d6530c99b979","last_reissued_at":"2026-05-18T01:07:24.590750Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:24.590750Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extensive Properties of the Complex Ginzburg-Landau Equation","license":"","headline":"","cross_cats":["nlin.CD","nlin.PS","patt-sol"],"primary_cat":"chao-dyn","authors_text":"Jean-Pierre Eckmann, Pierre Collet","submitted_at":"1998-02-06T07:25:42Z","abstract_excerpt":"We study the set of solutions of the complex Ginzburg-Landau equation in $\\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(\\epsilon)$ of balls of radius $\\epsilon$ in $\\Linfty(Q_L)$. We show that the Kolmogorov $\\epsilon$-entropy per unit length, $H_\\epsilon =\\lim_{L\\to\\infty} L^{-d} \\log N_{Q_L}(\\epsilon)$ exists. In particular, we bound $H_\\epsilon$ by $\\OO(\\log(1/\\epsilon)$, which shows that the attracting set is smaller than the set o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"chao-dyn/9802006","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"chao-dyn/9802006","created_at":"2026-05-18T01:07:24.590837+00:00"},{"alias_kind":"arxiv_version","alias_value":"chao-dyn/9802006v1","created_at":"2026-05-18T01:07:24.590837+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.chao-dyn/9802006","created_at":"2026-05-18T01:07:24.590837+00:00"},{"alias_kind":"pith_short_12","alias_value":"WG544QPMOWFN","created_at":"2026-05-18T12:25:49.038998+00:00"},{"alias_kind":"pith_short_16","alias_value":"WG544QPMOWFNRVM2","created_at":"2026-05-18T12:25:49.038998+00:00"},{"alias_kind":"pith_short_8","alias_value":"WG544QPM","created_at":"2026-05-18T12:25:49.038998+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E","json":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E.json","graph_json":"https://pith.science/api/pith-number/WG544QPMOWFNRVM2JIW6GGXH7E/graph.json","events_json":"https://pith.science/api/pith-number/WG544QPMOWFNRVM2JIW6GGXH7E/events.json","paper":"https://pith.science/paper/WG544QPM"},"agent_actions":{"view_html":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E","download_json":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E.json","view_paper":"https://pith.science/paper/WG544QPM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=chao-dyn/9802006&json=true","fetch_graph":"https://pith.science/api/pith-number/WG544QPMOWFNRVM2JIW6GGXH7E/graph.json","fetch_events":"https://pith.science/api/pith-number/WG544QPMOWFNRVM2JIW6GGXH7E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E/action/storage_attestation","attest_author":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E/action/author_attestation","sign_citation":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E/action/citation_signature","submit_replication":"https://pith.science/pith/WG544QPMOWFNRVM2JIW6GGXH7E/action/replication_record"}},"created_at":"2026-05-18T01:07:24.590837+00:00","updated_at":"2026-05-18T01:07:24.590837+00:00"}