{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:SCE63BZDGX4ZJQTGJIWB2SUJWP","short_pith_number":"pith:SCE63BZD","schema_version":"1.0","canonical_sha256":"9089ed872335f994c2664a2c1d4a89b3e758331ae70245159aa0c352f72fb650","source":{"kind":"arxiv","id":"1504.02355","version":1},"attestation_state":"computed","paper":{"title":"A $0-2$ law for cosine families with $\\limsup$ to $\\infty$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Felix L. Schwenninger, Hans Zwart","submitted_at":"2015-04-09T15:33:23Z","abstract_excerpt":"For $\\left(C(t)\\right)_{t\\in\\mathbb R}$ being a cosine family on a unital normed algebra, we show that the estimate $\\limsup_{t\\to\\infty^{+}}\\|C(t) - I\\| <2$ implies that $C(t)=I$ for all $t\\in\\mathbb R$. This generalizes the result that $\\sup_{t\\geq0}\\|C(t)-I\\|<2$ yields that $C(t)=I$ for all $t\\geq0$. We also state the corresponding result for discrete cosine families and for semigroups."},"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":"1504.02355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-09T15:33:23Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"60d15e5051df9ee9334027eaf8438593c446316e966faeffa538895139f129aa","abstract_canon_sha256":"9a22b74a7f2d5b7a6114de4cebc8d130028c657393188a24f9b36bc6cf48ca42"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:15.312125Z","signature_b64":"Y9cYz91l2mP/XqSMPYoZ4XxHe1g68cLR4BfPEOplpq6fMwlVe5/KLDKFMY4uPhDp/j5ibbwsyHlQQftaUYOIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9089ed872335f994c2664a2c1d4a89b3e758331ae70245159aa0c352f72fb650","last_reissued_at":"2026-05-18T02:19:15.311501Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:15.311501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A $0-2$ law for cosine families with $\\limsup$ to $\\infty$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Felix L. Schwenninger, Hans Zwart","submitted_at":"2015-04-09T15:33:23Z","abstract_excerpt":"For $\\left(C(t)\\right)_{t\\in\\mathbb R}$ being a cosine family on a unital normed algebra, we show that the estimate $\\limsup_{t\\to\\infty^{+}}\\|C(t) - I\\| <2$ implies that $C(t)=I$ for all $t\\in\\mathbb R$. This generalizes the result that $\\sup_{t\\geq0}\\|C(t)-I\\|<2$ yields that $C(t)=I$ for all $t\\geq0$. We also state the corresponding result for discrete cosine families and for semigroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02355","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":"1504.02355","created_at":"2026-05-18T02:19:15.311585+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.02355v1","created_at":"2026-05-18T02:19:15.311585+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02355","created_at":"2026-05-18T02:19:15.311585+00:00"},{"alias_kind":"pith_short_12","alias_value":"SCE63BZDGX4Z","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"SCE63BZDGX4ZJQTG","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"SCE63BZD","created_at":"2026-05-18T12:29:39.896362+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/SCE63BZDGX4ZJQTGJIWB2SUJWP","json":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP.json","graph_json":"https://pith.science/api/pith-number/SCE63BZDGX4ZJQTGJIWB2SUJWP/graph.json","events_json":"https://pith.science/api/pith-number/SCE63BZDGX4ZJQTGJIWB2SUJWP/events.json","paper":"https://pith.science/paper/SCE63BZD"},"agent_actions":{"view_html":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP","download_json":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP.json","view_paper":"https://pith.science/paper/SCE63BZD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.02355&json=true","fetch_graph":"https://pith.science/api/pith-number/SCE63BZDGX4ZJQTGJIWB2SUJWP/graph.json","fetch_events":"https://pith.science/api/pith-number/SCE63BZDGX4ZJQTGJIWB2SUJWP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP/action/storage_attestation","attest_author":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP/action/author_attestation","sign_citation":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP/action/citation_signature","submit_replication":"https://pith.science/pith/SCE63BZDGX4ZJQTGJIWB2SUJWP/action/replication_record"}},"created_at":"2026-05-18T02:19:15.311585+00:00","updated_at":"2026-05-18T02:19:15.311585+00:00"}