{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:LYEZHMV5TBLB5ZD2YW5QMBPEAJ","short_pith_number":"pith:LYEZHMV5","canonical_record":{"source":{"id":"1605.06984","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-23T11:54:08Z","cross_cats_sorted":[],"title_canon_sha256":"2a7a05f8646891d642c11904ca656681789f30b9b37940195e09d878a732ac86","abstract_canon_sha256":"94cee0f279af4cb88aa15431bf443b19691e3a4157968cb75e93bc303ca231f9"},"schema_version":"1.0"},"canonical_sha256":"5e0993b2bd98561ee47ac5bb0605e4024f7019abffe829211ef2c26cc7ad76fb","source":{"kind":"arxiv","id":"1605.06984","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06984","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06984v2","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06984","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"pith_short_12","alias_value":"LYEZHMV5TBLB","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LYEZHMV5TBLB5ZD2","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LYEZHMV5","created_at":"2026-05-18T12:30:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:LYEZHMV5TBLB5ZD2YW5QMBPEAJ","target":"record","payload":{"canonical_record":{"source":{"id":"1605.06984","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-23T11:54:08Z","cross_cats_sorted":[],"title_canon_sha256":"2a7a05f8646891d642c11904ca656681789f30b9b37940195e09d878a732ac86","abstract_canon_sha256":"94cee0f279af4cb88aa15431bf443b19691e3a4157968cb75e93bc303ca231f9"},"schema_version":"1.0"},"canonical_sha256":"5e0993b2bd98561ee47ac5bb0605e4024f7019abffe829211ef2c26cc7ad76fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:40.615392Z","signature_b64":"EdUANYzP4k0WTF+ARriyjFGug5vPGa0wsd7BtQnyF5EOvvXN9236rWoC34UaFhzKmA6Vmc/pNwItVflrtH+PCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e0993b2bd98561ee47ac5bb0605e4024f7019abffe829211ef2c26cc7ad76fb","last_reissued_at":"2026-05-18T01:06:40.614762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:40.614762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.06984","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I9o+Ycf+F3lZwA+SjZYfKUu67fMDiR66SPINo+lOV3ZlxF8GaXzflIw08c5r+H1mk4rl7redYNxxw6DLJUcSCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:26:33.505944Z"},"content_sha256":"577cd8be91b2ed0b065ea2ba3289a113fae86216f6860ea0263f9cc19ef5398c","schema_version":"1.0","event_id":"sha256:577cd8be91b2ed0b065ea2ba3289a113fae86216f6860ea0263f9cc19ef5398c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:LYEZHMV5TBLB5ZD2YW5QMBPEAJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inequalities on generalized matrix functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Chi-Kwong Li, Qing-Wen Wang, Shaowu Huang, Yiu-Tung Poon","submitted_at":"2016-05-23T11:54:08Z","abstract_excerpt":"We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \\in \\{1\\} \\cup [2, \\infty)$, positive semi-definite matrices $A_i,\\ B_i,\\ C_i\\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_\\chi, d_\\xi$ such as the determinant and permanent, etc., we have \\begin{eqnarray*}&&\\left(d_\\chi(A_1+B_1+C_1)d_\\xi(A_2+B_2+C_2)\\right)^r \\\\ &&\\hskip 1in\n  + \\left(d_\\chi(A_1)d_\\xi(A_2)\\right)^r\n  + \\left(d_\\chi(B_1)d_\\xi(B_2)\\right)^r + \\left(d_\\chi(C_1)d_\\xi(C_2)\\right)^r \\\\ & \\ge &\\left("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06984","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LU6JLS0akw0Cp+EKmXIZ1ahS+WAWw2L/dCO7ooz/Grc9o0gLhLOPylHrf8hfGC2W9IOFHKLFSvZ/s7Dw1ri8DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:26:33.506765Z"},"content_sha256":"60c80bfa8040dd49fb196bcfc48d142d898a028921210930e95e23c24bba70cf","schema_version":"1.0","event_id":"sha256:60c80bfa8040dd49fb196bcfc48d142d898a028921210930e95e23c24bba70cf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/bundle.json","state_url":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T19:26:33Z","links":{"resolver":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ","bundle":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/bundle.json","state":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LYEZHMV5TBLB5ZD2YW5QMBPEAJ","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":"94cee0f279af4cb88aa15431bf443b19691e3a4157968cb75e93bc303ca231f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-23T11:54:08Z","title_canon_sha256":"2a7a05f8646891d642c11904ca656681789f30b9b37940195e09d878a732ac86"},"schema_version":"1.0","source":{"id":"1605.06984","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06984","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06984v2","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06984","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"pith_short_12","alias_value":"LYEZHMV5TBLB","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LYEZHMV5TBLB5ZD2","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LYEZHMV5","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:60c80bfa8040dd49fb196bcfc48d142d898a028921210930e95e23c24bba70cf","target":"graph","created_at":"2026-05-18T01:06:40Z","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":"We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \\in \\{1\\} \\cup [2, \\infty)$, positive semi-definite matrices $A_i,\\ B_i,\\ C_i\\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_\\chi, d_\\xi$ such as the determinant and permanent, etc., we have \\begin{eqnarray*}&&\\left(d_\\chi(A_1+B_1+C_1)d_\\xi(A_2+B_2+C_2)\\right)^r \\\\ &&\\hskip 1in\n  + \\left(d_\\chi(A_1)d_\\xi(A_2)\\right)^r\n  + \\left(d_\\chi(B_1)d_\\xi(B_2)\\right)^r + \\left(d_\\chi(C_1)d_\\xi(C_2)\\right)^r \\\\ & \\ge &\\left(","authors_text":"Chi-Kwong Li, Qing-Wen Wang, Shaowu Huang, Yiu-Tung Poon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-23T11:54:08Z","title":"Inequalities on generalized matrix functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06984","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":"sha256:577cd8be91b2ed0b065ea2ba3289a113fae86216f6860ea0263f9cc19ef5398c","target":"record","created_at":"2026-05-18T01:06:40Z","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":"94cee0f279af4cb88aa15431bf443b19691e3a4157968cb75e93bc303ca231f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-23T11:54:08Z","title_canon_sha256":"2a7a05f8646891d642c11904ca656681789f30b9b37940195e09d878a732ac86"},"schema_version":"1.0","source":{"id":"1605.06984","kind":"arxiv","version":2}},"canonical_sha256":"5e0993b2bd98561ee47ac5bb0605e4024f7019abffe829211ef2c26cc7ad76fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5e0993b2bd98561ee47ac5bb0605e4024f7019abffe829211ef2c26cc7ad76fb","first_computed_at":"2026-05-18T01:06:40.614762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:40.614762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EdUANYzP4k0WTF+ARriyjFGug5vPGa0wsd7BtQnyF5EOvvXN9236rWoC34UaFhzKmA6Vmc/pNwItVflrtH+PCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:40.615392Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.06984","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:577cd8be91b2ed0b065ea2ba3289a113fae86216f6860ea0263f9cc19ef5398c","sha256:60c80bfa8040dd49fb196bcfc48d142d898a028921210930e95e23c24bba70cf"],"state_sha256":"1ff6029f3a2f61dec4be8a4efa7214e4dd7eb1c12e35086f490d96d8411b211f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZG+oc/go6OKM3pVI5tqJCqycX5/eThL6xJHbehc3T0UVlJM2clO4QrtTxLJfe6Wl/nGsULKLWwQFpHt7nmzNAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T19:26:33.512523Z","bundle_sha256":"9f8f29003e177a2f7dd0f98624064278748932bb460b02f2eda99195513664ec"}}