{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:VZLOTNEBDT44P5PCKBGCW3ZW7Q","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":"5af49e0f2bbe8be3df74cfb31fa26da96933e2be907deadd35f703585813d803","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:53:43Z","title_canon_sha256":"53c0fc203fbe152f969fceb69c42d05ea2c027c7ed19f13c75f71273ce2e8326"},"schema_version":"1.0","source":{"id":"1303.3979","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3979","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3979v1","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3979","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"pith_short_12","alias_value":"VZLOTNEBDT44","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VZLOTNEBDT44P5PC","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VZLOTNEB","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:41b19ea6843dc24fed635da14db6b18af683dbc6e155666439ba8f73c7c7b004","target":"graph","created_at":"2026-05-18T03:30:40Z","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":"In this article we examine the densities of a product and a ratio of two real positive definite matrix-variate random variables $X_1$ and $X_2$, which are statistically independently distributed, and we consider the density of the product $U_1=X_2^{1\\over2}X_1X_2^{1\\over2}$ as well as the density of the ratio $U_2=X_2^{1\\over2}X_1^{-1}X_2^{1\\over2}$. We define matrix-variate Kober fractional integral operators of the first and second kinds from a statistical perspective, making use of the derivation in the predecessor of this paper for the scalar variable case, by deriving the densities of pro","authors_text":"A.M. Mathai, H.J. Haubold","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:53:43Z","title":"Erdelyi-Kober Fractional Integral Operators from a Statistical Perspective -II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3979","kind":"arxiv","version":1},"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":"sha256:1860b769accaa136521af7d2566aee1b52ceda23d66c5a4cde47125f239c0bb6","target":"record","created_at":"2026-05-18T03:30:40Z","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":"5af49e0f2bbe8be3df74cfb31fa26da96933e2be907deadd35f703585813d803","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:53:43Z","title_canon_sha256":"53c0fc203fbe152f969fceb69c42d05ea2c027c7ed19f13c75f71273ce2e8326"},"schema_version":"1.0","source":{"id":"1303.3979","kind":"arxiv","version":1}},"canonical_sha256":"ae56e9b4811cf9c7f5e2504c2b6f36fc01cbfad58bd739a0320ee3971cbfee8a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ae56e9b4811cf9c7f5e2504c2b6f36fc01cbfad58bd739a0320ee3971cbfee8a","first_computed_at":"2026-05-18T03:30:40.296252Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:30:40.296252Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XW85Bp/Lqt/iaFBW8oM3Y9CUoxd+HBYFwUzFIEr3zgIu45zC8BGUExxY4drmjXkbEv0u9D0wurpu+1Aw01ZjDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:30:40.297197Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.3979","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1860b769accaa136521af7d2566aee1b52ceda23d66c5a4cde47125f239c0bb6","sha256:41b19ea6843dc24fed635da14db6b18af683dbc6e155666439ba8f73c7c7b004"],"state_sha256":"05282b33d2660e4a856ffa59bdaad23deccb1dde3ca324a574aa05aba952bb91"}