{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:Q7AFLH3XC3NU6JJX6FQBD4Y76T","short_pith_number":"pith:Q7AFLH3X","canonical_record":{"source":{"id":"1904.10783","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-24T13:07:42Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"161b81af28897a7b61e54b108444eccdeb95467e221cb957ac4cd741a83b1154","abstract_canon_sha256":"a9dcd968695447bb676840db390531748ac566d5e3e103fe6911be3b3aa2f322"},"schema_version":"1.0"},"canonical_sha256":"87c0559f7716db4f2537f16011f31ff4ee53d6e50af45c7173c9e4a403e72b0d","source":{"kind":"arxiv","id":"1904.10783","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.10783","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"1904.10783v2","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.10783","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"Q7AFLH3XC3NU","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"pith_short_16","alias_value":"Q7AFLH3XC3NU6JJX","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"pith_short_8","alias_value":"Q7AFLH3X","created_at":"2026-06-04T20:14:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:Q7AFLH3XC3NU6JJX6FQBD4Y76T","target":"record","payload":{"canonical_record":{"source":{"id":"1904.10783","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-24T13:07:42Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"161b81af28897a7b61e54b108444eccdeb95467e221cb957ac4cd741a83b1154","abstract_canon_sha256":"a9dcd968695447bb676840db390531748ac566d5e3e103fe6911be3b3aa2f322"},"schema_version":"1.0"},"canonical_sha256":"87c0559f7716db4f2537f16011f31ff4ee53d6e50af45c7173c9e4a403e72b0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T20:14:27.123624Z","signature_b64":"qciKnJlS3AE/1BoQssfmRC0eFFTfx1IzIQgsC/Qjzv9kwxd2tExoLDlPddkBP4v2313zIFXsxOGVp+DwyY2nAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87c0559f7716db4f2537f16011f31ff4ee53d6e50af45c7173c9e4a403e72b0d","last_reissued_at":"2026-06-04T20:14:27.123214Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T20:14:27.123214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.10783","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":"2026-06-04T20:14:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cae7Ib94xs3/nDkfCD8W8tVeG4KxwzM8thwqOZ0xJn/8QEObCCtGbaonOeQtL2S1Ur8gddv2UmwocbrpvUowDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:10:20.680083Z"},"content_sha256":"c82c25b9e0e4af10f47024302c286f0ec5435febe76504dae8a60f711871f054","schema_version":"1.0","event_id":"sha256:c82c25b9e0e4af10f47024302c286f0ec5435febe76504dae8a60f711871f054"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:Q7AFLH3XC3NU6JJX6FQBD4Y76T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Further results on the Drazin inverse of even-order tensors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Ashish Kumar Nandi, Jajati Keshari Sahoo, Ratikanta Behera","submitted_at":"2019-04-24T13:07:42Z","abstract_excerpt":"The notion of the Drazin inverse of an even-order tensor with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by producing a few characterizations of the Drazin inverse and the W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution to the multilinear systems using the Drazin inverse and iterative (higher order Gau"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.10783","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1904.10783/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":"2026-06-04T20:14:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WUkIS5vxsDEjSkknrhOncZjyJ/CxA2vcWbyYH14AUNpZzgw5sRfmraYtYCDPMboyW3MKMd/ElwCObEn6kFjRDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:10:20.680491Z"},"content_sha256":"3dc567749598fe63eb78a3031368628776185586f48b0e5d3292eee6acf8c4b1","schema_version":"1.0","event_id":"sha256:3dc567749598fe63eb78a3031368628776185586f48b0e5d3292eee6acf8c4b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T/bundle.json","state_url":"https://pith.science/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T/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":"2026-06-24T20:10:20Z","links":{"resolver":"https://pith.science/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T","bundle":"https://pith.science/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T/bundle.json","state":"https://pith.science/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q7AFLH3XC3NU6JJX6FQBD4Y76T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:Q7AFLH3XC3NU6JJX6FQBD4Y76T","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":"a9dcd968695447bb676840db390531748ac566d5e3e103fe6911be3b3aa2f322","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-24T13:07:42Z","title_canon_sha256":"161b81af28897a7b61e54b108444eccdeb95467e221cb957ac4cd741a83b1154"},"schema_version":"1.0","source":{"id":"1904.10783","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.10783","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"1904.10783v2","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.10783","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"Q7AFLH3XC3NU","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"pith_short_16","alias_value":"Q7AFLH3XC3NU6JJX","created_at":"2026-06-04T20:14:27Z"},{"alias_kind":"pith_short_8","alias_value":"Q7AFLH3X","created_at":"2026-06-04T20:14:27Z"}],"graph_snapshots":[{"event_id":"sha256:3dc567749598fe63eb78a3031368628776185586f48b0e5d3292eee6acf8c4b1","target":"graph","created_at":"2026-06-04T20:14:27Z","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/1904.10783/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The notion of the Drazin inverse of an even-order tensor with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by producing a few characterizations of the Drazin inverse and the W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution to the multilinear systems using the Drazin inverse and iterative (higher order Gau","authors_text":"Ashish Kumar Nandi, Jajati Keshari Sahoo, Ratikanta Behera","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-24T13:07:42Z","title":"Further results on the Drazin inverse of even-order tensors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.10783","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:c82c25b9e0e4af10f47024302c286f0ec5435febe76504dae8a60f711871f054","target":"record","created_at":"2026-06-04T20:14:27Z","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":"a9dcd968695447bb676840db390531748ac566d5e3e103fe6911be3b3aa2f322","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-24T13:07:42Z","title_canon_sha256":"161b81af28897a7b61e54b108444eccdeb95467e221cb957ac4cd741a83b1154"},"schema_version":"1.0","source":{"id":"1904.10783","kind":"arxiv","version":2}},"canonical_sha256":"87c0559f7716db4f2537f16011f31ff4ee53d6e50af45c7173c9e4a403e72b0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87c0559f7716db4f2537f16011f31ff4ee53d6e50af45c7173c9e4a403e72b0d","first_computed_at":"2026-06-04T20:14:27.123214Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T20:14:27.123214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qciKnJlS3AE/1BoQssfmRC0eFFTfx1IzIQgsC/Qjzv9kwxd2tExoLDlPddkBP4v2313zIFXsxOGVp+DwyY2nAQ==","signature_status":"signed_v1","signed_at":"2026-06-04T20:14:27.123624Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.10783","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c82c25b9e0e4af10f47024302c286f0ec5435febe76504dae8a60f711871f054","sha256:3dc567749598fe63eb78a3031368628776185586f48b0e5d3292eee6acf8c4b1"],"state_sha256":"b7c0cd661dd2e013ba2039291da8bdd5ff9bfc28a2e95c6a52410a994714ac48"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yHRfGxithEwSg2PS/aVANOEhNVPGlCRubl+GnjflvIk5HeJrb+ZqfLZiqfBF5Hjs8WSrgQskzqadRs0n+VfdAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T20:10:20.682530Z","bundle_sha256":"673814e0e12a2e96171bf0abbff15f7f70700d00190036c47f5ca11ca6e58544"}}