{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:BEXR2BCAAJIXSET3TADE46EY55","short_pith_number":"pith:BEXR2BCA","canonical_record":{"source":{"id":"1908.01131","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2019-08-03T07:58:19Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"b9c0ba5c1f167ca71fe83f7d041f1fcb63c1b46349bb29d14fb3407fb6b31b46","abstract_canon_sha256":"44369f99bfa94b44d3db0734a58a58862b4ce3ccfa642c90df87aa209149ad11"},"schema_version":"1.0"},"canonical_sha256":"092f1d0440025179127b98064e7898ef404fae1d9c7b5619343cbc246bb3fb07","source":{"kind":"arxiv","id":"1908.01131","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01131","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01131v3","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01131","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"pith_short_12","alias_value":"BEXR2BCAAJIX","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"pith_short_16","alias_value":"BEXR2BCAAJIXSET3","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"pith_short_8","alias_value":"BEXR2BCA","created_at":"2026-07-05T05:23:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:BEXR2BCAAJIXSET3TADE46EY55","target":"record","payload":{"canonical_record":{"source":{"id":"1908.01131","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2019-08-03T07:58:19Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"b9c0ba5c1f167ca71fe83f7d041f1fcb63c1b46349bb29d14fb3407fb6b31b46","abstract_canon_sha256":"44369f99bfa94b44d3db0734a58a58862b4ce3ccfa642c90df87aa209149ad11"},"schema_version":"1.0"},"canonical_sha256":"092f1d0440025179127b98064e7898ef404fae1d9c7b5619343cbc246bb3fb07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:23:27.652498Z","signature_b64":"SbyKE1JzGH905yNB66WBeMCjO7LxmcR3QVdK9yOTYxeJd5/GCo5ADjAffW2zcjbVkL/XbgRbMk2EUGxw83j8Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"092f1d0440025179127b98064e7898ef404fae1d9c7b5619343cbc246bb3fb07","last_reissued_at":"2026-07-05T05:23:27.652096Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:23:27.652096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.01131","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:23:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iqBj3Qxx7d4PMnDxWdk7TYNXcmC+rXNhj125k/KoM+zPETSvgncNTFyJ6XIOanPltEXXX6FL0Fca/vIA8esLAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:56:58.033594Z"},"content_sha256":"f8fb6511b2c3fde395f5972b053c89037f9642c8f94c6f4bdde8dfa16ece4f92","schema_version":"1.0","event_id":"sha256:f8fb6511b2c3fde395f5972b053c89037f9642c8f94c6f4bdde8dfa16ece4f92"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:BEXR2BCAAJIXSET3TADE46EY55","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random Tensors and their Normal Distributions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Changqing Xu, Kaijie Xu","submitted_at":"2019-08-03T07:58:19Z","abstract_excerpt":"The main purpose of this paper is to introduce the random tensor with normal distribution, which promotes the matrix normal distribution to a higher order case. Some basic knowledge on tensors are introduced before we focus on the random tensors whose entries follow normal distribution. The random tensor with standard normal distribution(SND) is introduced as an extension of random normal matrices. As a random multi-array deduced from an affine transformation on a SND tensor, the general normal random tensor is initialised in the paper. We then investigate some equivalent definitions of a norm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01131","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01131/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T05:23:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TjwdAKY9G5I8qCiD8daNf86t7L0aOLdO3YNGi4D/zglM5R6PGTnY5OL/jmhetHvv2f3I7zS1o6hOKDYBZboQBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:56:58.034448Z"},"content_sha256":"5aee22fb1d139a3e62acccc7375f8717ceef75295e0f1b8183238b927c349494","schema_version":"1.0","event_id":"sha256:5aee22fb1d139a3e62acccc7375f8717ceef75295e0f1b8183238b927c349494"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BEXR2BCAAJIXSET3TADE46EY55/bundle.json","state_url":"https://pith.science/pith/BEXR2BCAAJIXSET3TADE46EY55/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BEXR2BCAAJIXSET3TADE46EY55/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":"2026-08-16T08:56:58Z","links":{"resolver":"https://pith.science/pith/BEXR2BCAAJIXSET3TADE46EY55","bundle":"https://pith.science/pith/BEXR2BCAAJIXSET3TADE46EY55/bundle.json","state":"https://pith.science/pith/BEXR2BCAAJIXSET3TADE46EY55/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BEXR2BCAAJIXSET3TADE46EY55/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:BEXR2BCAAJIXSET3TADE46EY55","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":"44369f99bfa94b44d3db0734a58a58862b4ce3ccfa642c90df87aa209149ad11","cross_cats_sorted":["stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2019-08-03T07:58:19Z","title_canon_sha256":"b9c0ba5c1f167ca71fe83f7d041f1fcb63c1b46349bb29d14fb3407fb6b31b46"},"schema_version":"1.0","source":{"id":"1908.01131","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01131","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01131v3","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01131","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"pith_short_12","alias_value":"BEXR2BCAAJIX","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"pith_short_16","alias_value":"BEXR2BCAAJIXSET3","created_at":"2026-07-05T05:23:27Z"},{"alias_kind":"pith_short_8","alias_value":"BEXR2BCA","created_at":"2026-07-05T05:23:27Z"}],"graph_snapshots":[{"event_id":"sha256:5aee22fb1d139a3e62acccc7375f8717ceef75295e0f1b8183238b927c349494","target":"graph","created_at":"2026-07-05T05:23:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.01131/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The main purpose of this paper is to introduce the random tensor with normal distribution, which promotes the matrix normal distribution to a higher order case. Some basic knowledge on tensors are introduced before we focus on the random tensors whose entries follow normal distribution. The random tensor with standard normal distribution(SND) is introduced as an extension of random normal matrices. As a random multi-array deduced from an affine transformation on a SND tensor, the general normal random tensor is initialised in the paper. We then investigate some equivalent definitions of a norm","authors_text":"Changqing Xu, Kaijie Xu","cross_cats":["stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2019-08-03T07:58:19Z","title":"Random Tensors and their Normal Distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01131","kind":"arxiv","version":3},"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:f8fb6511b2c3fde395f5972b053c89037f9642c8f94c6f4bdde8dfa16ece4f92","target":"record","created_at":"2026-07-05T05:23:27Z","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":"44369f99bfa94b44d3db0734a58a58862b4ce3ccfa642c90df87aa209149ad11","cross_cats_sorted":["stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2019-08-03T07:58:19Z","title_canon_sha256":"b9c0ba5c1f167ca71fe83f7d041f1fcb63c1b46349bb29d14fb3407fb6b31b46"},"schema_version":"1.0","source":{"id":"1908.01131","kind":"arxiv","version":3}},"canonical_sha256":"092f1d0440025179127b98064e7898ef404fae1d9c7b5619343cbc246bb3fb07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"092f1d0440025179127b98064e7898ef404fae1d9c7b5619343cbc246bb3fb07","first_computed_at":"2026-07-05T05:23:27.652096Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:23:27.652096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SbyKE1JzGH905yNB66WBeMCjO7LxmcR3QVdK9yOTYxeJd5/GCo5ADjAffW2zcjbVkL/XbgRbMk2EUGxw83j8Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T05:23:27.652498Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.01131","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f8fb6511b2c3fde395f5972b053c89037f9642c8f94c6f4bdde8dfa16ece4f92","sha256:5aee22fb1d139a3e62acccc7375f8717ceef75295e0f1b8183238b927c349494"],"state_sha256":"784d5c301862cad4f1f3c7789aee5cfc0ebaf87ee7f29e9c5e1aab8e8ca0cddd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HPLX4eK2FAZDSiTPEr7mokvt8dptaWb7YiTUcc6NfmQC+v53u5wsNGtJ4CcQ/UWERGZjEQWH9CS2bakb0MOBAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T08:56:58.040670Z","bundle_sha256":"5099fa1cf8d1e4e7c90758366ca9c80e517ef0d051d31159573646db65e941bc"}}