{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:WN7HBIF5OBBNQO4D7HNA2WEMI3","short_pith_number":"pith:WN7HBIF5","schema_version":"1.0","canonical_sha256":"b37e70a0bd7042d83b83f9da0d588c46fd033bc9e3fcb527cf4653ed6bf984ab","source":{"kind":"arxiv","id":"1705.01327","version":1},"attestation_state":"computed","paper":{"title":"The geometrical origins of some distributions and the complete concentration of measure phenomenon for mean-values of functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Cheng-shi Liu","submitted_at":"2017-05-03T09:35:38Z","abstract_excerpt":"We derive out naturally some important distributions such as high order normal distributions and high order exponent distributions and the Gamma distribution from a geometrical way. Further, we obtain the exact mean-values of integral form functionals in the balls of continuous functions space with $p-$norm, and show the complete concentration of measure phenomenon which means that a functional takes its average on a ball with probability 1, from which we have nonlinear exchange formula of expectation."},"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":"1705.01327","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-05-03T09:35:38Z","cross_cats_sorted":["math-ph","math.FA","math.MP","math.ST","stat.TH"],"title_canon_sha256":"d946306231a44ad3929083967111c52f41e795c42f021c9a2a1654847685faf1","abstract_canon_sha256":"bc6e6b380411d468952a26099b2bc582575df535154a4e7925ec28983ae30e7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:05.425741Z","signature_b64":"RZnVk3Jdt4uIaNG5D6aTEbXCEwU0jy9CCD97xmEhK7dW/FYkFkpX5xj1zW4T9wilf2DXFNVyoZvynRn1uIOMBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b37e70a0bd7042d83b83f9da0d588c46fd033bc9e3fcb527cf4653ed6bf984ab","last_reissued_at":"2026-05-18T00:45:05.425169Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:05.425169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The geometrical origins of some distributions and the complete concentration of measure phenomenon for mean-values of functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Cheng-shi Liu","submitted_at":"2017-05-03T09:35:38Z","abstract_excerpt":"We derive out naturally some important distributions such as high order normal distributions and high order exponent distributions and the Gamma distribution from a geometrical way. Further, we obtain the exact mean-values of integral form functionals in the balls of continuous functions space with $p-$norm, and show the complete concentration of measure phenomenon which means that a functional takes its average on a ball with probability 1, from which we have nonlinear exchange formula of expectation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01327","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":"1705.01327","created_at":"2026-05-18T00:45:05.425262+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.01327v1","created_at":"2026-05-18T00:45:05.425262+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.01327","created_at":"2026-05-18T00:45:05.425262+00:00"},{"alias_kind":"pith_short_12","alias_value":"WN7HBIF5OBBN","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WN7HBIF5OBBNQO4D","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WN7HBIF5","created_at":"2026-05-18T12:31:53.515858+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/WN7HBIF5OBBNQO4D7HNA2WEMI3","json":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3.json","graph_json":"https://pith.science/api/pith-number/WN7HBIF5OBBNQO4D7HNA2WEMI3/graph.json","events_json":"https://pith.science/api/pith-number/WN7HBIF5OBBNQO4D7HNA2WEMI3/events.json","paper":"https://pith.science/paper/WN7HBIF5"},"agent_actions":{"view_html":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3","download_json":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3.json","view_paper":"https://pith.science/paper/WN7HBIF5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.01327&json=true","fetch_graph":"https://pith.science/api/pith-number/WN7HBIF5OBBNQO4D7HNA2WEMI3/graph.json","fetch_events":"https://pith.science/api/pith-number/WN7HBIF5OBBNQO4D7HNA2WEMI3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3/action/storage_attestation","attest_author":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3/action/author_attestation","sign_citation":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3/action/citation_signature","submit_replication":"https://pith.science/pith/WN7HBIF5OBBNQO4D7HNA2WEMI3/action/replication_record"}},"created_at":"2026-05-18T00:45:05.425262+00:00","updated_at":"2026-05-18T00:45:05.425262+00:00"}