{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PGDF3EJIWOXPFT53N7RISI4LI2","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":"fa6de7157a71584b861ae682cde83cbb51d423f1321c33b08a82d19c5ebb11eb","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-27T06:48:04Z","title_canon_sha256":"a37db37358d51fd34299e3310b34cf894e8dd8584729938744e20b2b8c25af52"},"schema_version":"1.0","source":{"id":"1207.6464","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6464","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6464v1","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6464","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"pith_short_12","alias_value":"PGDF3EJIWOXP","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PGDF3EJIWOXPFT53","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PGDF3EJI","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:e492e0c5694262bc779453b056b32a9b30286f514edb3a6371e9797f327d3a2c","target":"graph","created_at":"2026-05-18T03:49:59Z","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":"We study the total positivity of the multiplicative convolution kernel T associated with the independent product of two random variables $B(a,b)$ and $\\Gamma(c).$ This kernel is totally positive of infinite order if $b$ or $d = a+b -c$ are integers. Otherwise the sign-regularity of T has always a finite order, which is here computed. More precisely, for every $n\\ge 1$ it is shown that T is totally positive of order $n + 1$ if and only if $(d,b)$ lies above a certain stairway ${\\mathcal E}_n$ plotted in the upper half-plane. This stairway also characterizes the sign-invariance of several determ","authors_text":"Thomas Simon (LPP)","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-27T06:48:04Z","title":"Produit Beta-Gamma et r\\'egularit\\'e du signe"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6464","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":"sha256:54423c9ffe8e7229e92a10f38d8d2355097734726b1f7b3be327b5e45a07b7df","target":"record","created_at":"2026-05-18T03:49:59Z","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":"fa6de7157a71584b861ae682cde83cbb51d423f1321c33b08a82d19c5ebb11eb","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-27T06:48:04Z","title_canon_sha256":"a37db37358d51fd34299e3310b34cf894e8dd8584729938744e20b2b8c25af52"},"schema_version":"1.0","source":{"id":"1207.6464","kind":"arxiv","version":1}},"canonical_sha256":"79865d9128b3aef2cfbb6fe289238b469a9104c540d4b055c3dcbb53cde013a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"79865d9128b3aef2cfbb6fe289238b469a9104c540d4b055c3dcbb53cde013a0","first_computed_at":"2026-05-18T03:49:59.263839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:59.263839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ybUiqd+pz7K3qcO1eGV86oPvf7IysN+8N6qDmF8AbZ4Y4fXMwmiFIPvxVBFolJgE8R91hxqYSjkybHVR9V7oDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:59.264477Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6464","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:54423c9ffe8e7229e92a10f38d8d2355097734726b1f7b3be327b5e45a07b7df","sha256:e492e0c5694262bc779453b056b32a9b30286f514edb3a6371e9797f327d3a2c"],"state_sha256":"12dec5258b8321ed0d124feeb9219cafa65be97c8440c6305a4dc367aaa5ecc0"}