{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:BPJ2RPHU5IR6SC2OXR5VFOCY2F","short_pith_number":"pith:BPJ2RPHU","schema_version":"1.0","canonical_sha256":"0bd3a8bcf4ea23e90b4ebc7b52b858d16a5a25c607e850956f22985650a83538","source":{"kind":"arxiv","id":"0712.2816","version":3},"attestation_state":"computed","paper":{"title":"Coverage processes on spheres and condition numbers for linear programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.PR","authors_text":"Felipe Cucker, Martin Lotz, Peter B\\\"urgisser","submitted_at":"2007-12-17T20:58:49Z","abstract_excerpt":"This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\\alpha)$ be the probability that $n$ spherical caps of angular radius $\\alpha$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\\alpha)$ in the case $\\alpha\\in[\\pi/2,\\pi]$ and an upper bound for $p(n,m,\\alpha)$ in the case $\\alpha\\in [0,\\pi/2]$ which tends to $p(n,m,\\pi/2)$ when $\\alpha\\to\\pi/2$. In the case $\\alpha\\in[0,\\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\\alpha$ that are needed to cover $S^m$. Secondly, we study the "},"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":"0712.2816","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-12-17T20:58:49Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"f4cb63a2a98a4672a7af6e9a2c6c14c8a625b02c9289c4021f01ee60f83571af","abstract_canon_sha256":"9e9266fd8784ca71daf2b58bf6341bbad05e05131b92950f854cd44150d18241"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:54.968189Z","signature_b64":"3tw+0NaWDLrwAxvEV5g8iMhAD04OsYw5hZWpInj8QrCk1ndBhK7E2yGnQZ7szFO3SBX23TCM4z4RLzLC1SLvDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bd3a8bcf4ea23e90b4ebc7b52b858d16a5a25c607e850956f22985650a83538","last_reissued_at":"2026-05-18T04:19:54.967644Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:54.967644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coverage processes on spheres and condition numbers for linear programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.PR","authors_text":"Felipe Cucker, Martin Lotz, Peter B\\\"urgisser","submitted_at":"2007-12-17T20:58:49Z","abstract_excerpt":"This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\\alpha)$ be the probability that $n$ spherical caps of angular radius $\\alpha$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\\alpha)$ in the case $\\alpha\\in[\\pi/2,\\pi]$ and an upper bound for $p(n,m,\\alpha)$ in the case $\\alpha\\in [0,\\pi/2]$ which tends to $p(n,m,\\pi/2)$ when $\\alpha\\to\\pi/2$. In the case $\\alpha\\in[0,\\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\\alpha$ that are needed to cover $S^m$. Secondly, we study the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2816","kind":"arxiv","version":3},"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":"0712.2816","created_at":"2026-05-18T04:19:54.967728+00:00"},{"alias_kind":"arxiv_version","alias_value":"0712.2816v3","created_at":"2026-05-18T04:19:54.967728+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2816","created_at":"2026-05-18T04:19:54.967728+00:00"},{"alias_kind":"pith_short_12","alias_value":"BPJ2RPHU5IR6","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"BPJ2RPHU5IR6SC2O","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"BPJ2RPHU","created_at":"2026-05-18T12:25:55.427421+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/BPJ2RPHU5IR6SC2OXR5VFOCY2F","json":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F.json","graph_json":"https://pith.science/api/pith-number/BPJ2RPHU5IR6SC2OXR5VFOCY2F/graph.json","events_json":"https://pith.science/api/pith-number/BPJ2RPHU5IR6SC2OXR5VFOCY2F/events.json","paper":"https://pith.science/paper/BPJ2RPHU"},"agent_actions":{"view_html":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F","download_json":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F.json","view_paper":"https://pith.science/paper/BPJ2RPHU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0712.2816&json=true","fetch_graph":"https://pith.science/api/pith-number/BPJ2RPHU5IR6SC2OXR5VFOCY2F/graph.json","fetch_events":"https://pith.science/api/pith-number/BPJ2RPHU5IR6SC2OXR5VFOCY2F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F/action/storage_attestation","attest_author":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F/action/author_attestation","sign_citation":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F/action/citation_signature","submit_replication":"https://pith.science/pith/BPJ2RPHU5IR6SC2OXR5VFOCY2F/action/replication_record"}},"created_at":"2026-05-18T04:19:54.967728+00:00","updated_at":"2026-05-18T04:19:54.967728+00:00"}