{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:UJYVRYB4VAZUDQPIRKUX5K373L","short_pith_number":"pith:UJYVRYB4","canonical_record":{"source":{"id":"0806.4048","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2008-06-25T08:43:43Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"739567611a16bcc76ff43b6d90c50ae267e10fb2efd4920a6b8b9bcf6dabbef1","abstract_canon_sha256":"fcb64b873e81836af09823fccec1f1403a029f7e90531bb8bf03a2b93af43f22"},"schema_version":"1.0"},"canonical_sha256":"a27158e03ca83341c1e88aa97eab7fdaf54d7be5cad28f1374a3630278c98e7f","source":{"kind":"arxiv","id":"0806.4048","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0806.4048","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"0806.4048v2","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.4048","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"UJYVRYB4VAZU","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"UJYVRYB4VAZUDQPI","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"UJYVRYB4","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:UJYVRYB4VAZUDQPIRKUX5K373L","target":"record","payload":{"canonical_record":{"source":{"id":"0806.4048","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2008-06-25T08:43:43Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"739567611a16bcc76ff43b6d90c50ae267e10fb2efd4920a6b8b9bcf6dabbef1","abstract_canon_sha256":"fcb64b873e81836af09823fccec1f1403a029f7e90531bb8bf03a2b93af43f22"},"schema_version":"1.0"},"canonical_sha256":"a27158e03ca83341c1e88aa97eab7fdaf54d7be5cad28f1374a3630278c98e7f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:45.240148Z","signature_b64":"vW57pkg9vaV2sTDqD9Ju7M6f2fap+SSPtKqiMQBta52EoZSmMpAUGg7JgrVc5OH2T4Gss6x+4Od+DNLhF7Q6Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a27158e03ca83341c1e88aa97eab7fdaf54d7be5cad28f1374a3630278c98e7f","last_reissued_at":"2026-05-18T04:14:45.239718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:45.239718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0806.4048","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":"2026-05-18T04:14:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HxzJrr2ltC2eOo5yO/A1ONxBfH+xAlHMyu7Jb81LEfUKFGLd393474zOmde4f7FRLXmG+tN8aYYOVtFN7yLKCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T07:51:39.096169Z"},"content_sha256":"136526846b94f55f73de7c8a6f20657edfcc32e020b8390df8dedf89ef053a22","schema_version":"1.0","event_id":"sha256:136526846b94f55f73de7c8a6f20657edfcc32e020b8390df8dedf89ef053a22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:UJYVRYB4VAZUDQPIRKUX5K373L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"About the maximal rank of 3-tensors over the real and the complex number field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.RA","authors_text":"Mitsuhiro Miyazaki, Toshio Sakata, Toshio Sumi","submitted_at":"2008-06-25T08:43:43Z","abstract_excerpt":"High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.4048","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":"2026-05-18T04:14:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bOeq31JQLk4q34jH3DNCd0vCdC7QNrlIw3ZmB0MLlWZYGgCndEuZE+tXLijnZu1m9XCSgpCGmPfCUIzkCmpQBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T07:51:39.096807Z"},"content_sha256":"7f6fcc688f1961788e7c991c01350ce5f1a12cbabed77159c19138bab1470b41","schema_version":"1.0","event_id":"sha256:7f6fcc688f1961788e7c991c01350ce5f1a12cbabed77159c19138bab1470b41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UJYVRYB4VAZUDQPIRKUX5K373L/bundle.json","state_url":"https://pith.science/pith/UJYVRYB4VAZUDQPIRKUX5K373L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UJYVRYB4VAZUDQPIRKUX5K373L/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":"2026-06-24T07:51:39Z","links":{"resolver":"https://pith.science/pith/UJYVRYB4VAZUDQPIRKUX5K373L","bundle":"https://pith.science/pith/UJYVRYB4VAZUDQPIRKUX5K373L/bundle.json","state":"https://pith.science/pith/UJYVRYB4VAZUDQPIRKUX5K373L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UJYVRYB4VAZUDQPIRKUX5K373L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:UJYVRYB4VAZUDQPIRKUX5K373L","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":"fcb64b873e81836af09823fccec1f1403a029f7e90531bb8bf03a2b93af43f22","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2008-06-25T08:43:43Z","title_canon_sha256":"739567611a16bcc76ff43b6d90c50ae267e10fb2efd4920a6b8b9bcf6dabbef1"},"schema_version":"1.0","source":{"id":"0806.4048","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0806.4048","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"0806.4048v2","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.4048","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"UJYVRYB4VAZU","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"UJYVRYB4VAZUDQPI","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"UJYVRYB4","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:7f6fcc688f1961788e7c991c01350ce5f1a12cbabed77159c19138bab1470b41","target":"graph","created_at":"2026-05-18T04:14:45Z","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":"High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results.","authors_text":"Mitsuhiro Miyazaki, Toshio Sakata, Toshio Sumi","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2008-06-25T08:43:43Z","title":"About the maximal rank of 3-tensors over the real and the complex number field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.4048","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":"sha256:136526846b94f55f73de7c8a6f20657edfcc32e020b8390df8dedf89ef053a22","target":"record","created_at":"2026-05-18T04:14:45Z","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":"fcb64b873e81836af09823fccec1f1403a029f7e90531bb8bf03a2b93af43f22","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2008-06-25T08:43:43Z","title_canon_sha256":"739567611a16bcc76ff43b6d90c50ae267e10fb2efd4920a6b8b9bcf6dabbef1"},"schema_version":"1.0","source":{"id":"0806.4048","kind":"arxiv","version":2}},"canonical_sha256":"a27158e03ca83341c1e88aa97eab7fdaf54d7be5cad28f1374a3630278c98e7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a27158e03ca83341c1e88aa97eab7fdaf54d7be5cad28f1374a3630278c98e7f","first_computed_at":"2026-05-18T04:14:45.239718Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:45.239718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vW57pkg9vaV2sTDqD9Ju7M6f2fap+SSPtKqiMQBta52EoZSmMpAUGg7JgrVc5OH2T4Gss6x+4Od+DNLhF7Q6Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:45.240148Z","signed_message":"canonical_sha256_bytes"},"source_id":"0806.4048","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:136526846b94f55f73de7c8a6f20657edfcc32e020b8390df8dedf89ef053a22","sha256:7f6fcc688f1961788e7c991c01350ce5f1a12cbabed77159c19138bab1470b41"],"state_sha256":"f3c64973c3cf79566d78d580d7b4cc40515e6c055612d41ef35ea87710cd904e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Nb3qYqcM5t95cnj00MgU/kiNmqpeYnf74rkZGyxDjSVa6R3T6HFTyM44BeOpqDysvH8W8+VKMvh9McZEzt2Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T07:51:39.100238Z","bundle_sha256":"9c0f068a2ee0ee8bdda3ad45a7b9de573d3908292dc67c5a92f80db3ca83f21d"}}