{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:EG2RAKIL7ERO24MVDD2IIK7RFK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9c2fbb8befd0c255e762f21f919a2b22ffa35ecbc7f3648905ff2817c5553b56","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-21T19:57:39Z","title_canon_sha256":"a6f927c2b051313a14186360894696120cf5e33aa0c27ef823c1c452b487c1b4"},"schema_version":"1.0","source":{"id":"1505.05867","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.05867","created_at":"2026-05-18T01:38:53Z"},{"alias_kind":"arxiv_version","alias_value":"1505.05867v2","created_at":"2026-05-18T01:38:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.05867","created_at":"2026-05-18T01:38:53Z"},{"alias_kind":"pith_short_12","alias_value":"EG2RAKIL7ERO","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EG2RAKIL7ERO24MV","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EG2RAKIL","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:69a9f60856f22a904168885fb2d43e889f710cf2c608dfb325b97565603a4003","target":"graph","created_at":"2026-05-18T01:38:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The volume of the unit ball of the Lebesgue sequence space $\\ell_p^m$ is very well known since the times of Dirichlet. We calculate the volume of the unit ball in the mixed norm $\\ell^n_q(\\ell_p^m)$, whose special cases are nowadays popular in machine learning under the name of group lasso. We consider the real as well as the complex case. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We close by an overview of open problems.","authors_text":"Henning Kempka, Jan Vyb\\'iral","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-21T19:57:39Z","title":"Volumes of unit balls of mixed sequence spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.05867","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:58634a2a5ca3eb5adaed6e9648b00dcf930a15df7edb6b729d2dbd67a692755f","target":"record","created_at":"2026-05-18T01:38:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9c2fbb8befd0c255e762f21f919a2b22ffa35ecbc7f3648905ff2817c5553b56","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-21T19:57:39Z","title_canon_sha256":"a6f927c2b051313a14186360894696120cf5e33aa0c27ef823c1c452b487c1b4"},"schema_version":"1.0","source":{"id":"1505.05867","kind":"arxiv","version":2}},"canonical_sha256":"21b510290bf922ed719518f4842bf12a9293cd03414099ea5a0864a8dc2f9871","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21b510290bf922ed719518f4842bf12a9293cd03414099ea5a0864a8dc2f9871","first_computed_at":"2026-05-18T01:38:53.553551Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:53.553551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gtcJ9axdlRRpSmrejokiZBidsjS07OpEAqwh2Yr89Um8iDPTlPNS+lFf010rUDGUfddWQIOivn84T1Hcx6DJCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:53.554337Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.05867","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:58634a2a5ca3eb5adaed6e9648b00dcf930a15df7edb6b729d2dbd67a692755f","sha256:69a9f60856f22a904168885fb2d43e889f710cf2c608dfb325b97565603a4003"],"state_sha256":"7522f07a7147a3acff9f60e5f2169635cf67a9bb781bb5aad266b4a6619906c0"}