{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:K74CR3RCOCTR7TWENTFZJC3PMI","short_pith_number":"pith:K74CR3RC","canonical_record":{"source":{"id":"1907.08408","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2019-07-19T08:43:39Z","cross_cats_sorted":[],"title_canon_sha256":"2f13edab83d3ac22fe28e6648e27c1cdc6e73527d51045623ba6e89396f86c5a","abstract_canon_sha256":"f3bcde368ca8cfd2618623f2c8f5bc823a8325ddfc71594186351ded4814aa51"},"schema_version":"1.0"},"canonical_sha256":"57f828ee2270a71fcec46ccb948b6f6217a44cd3fb0d60f662db4f4ebca14f12","source":{"kind":"arxiv","id":"1907.08408","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.08408","created_at":"2026-05-17T23:40:09Z"},{"alias_kind":"arxiv_version","alias_value":"1907.08408v1","created_at":"2026-05-17T23:40:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08408","created_at":"2026-05-17T23:40:09Z"},{"alias_kind":"pith_short_12","alias_value":"K74CR3RCOCTR","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"K74CR3RCOCTR7TWE","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"K74CR3RC","created_at":"2026-05-18T12:33:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:K74CR3RCOCTR7TWENTFZJC3PMI","target":"record","payload":{"canonical_record":{"source":{"id":"1907.08408","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2019-07-19T08:43:39Z","cross_cats_sorted":[],"title_canon_sha256":"2f13edab83d3ac22fe28e6648e27c1cdc6e73527d51045623ba6e89396f86c5a","abstract_canon_sha256":"f3bcde368ca8cfd2618623f2c8f5bc823a8325ddfc71594186351ded4814aa51"},"schema_version":"1.0"},"canonical_sha256":"57f828ee2270a71fcec46ccb948b6f6217a44cd3fb0d60f662db4f4ebca14f12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:09.942437Z","signature_b64":"Y26W8pWhQ9rAvN6numCqC+FWj8rBsk2uXER2s2dLe9BfGbewxmjtgrpdxHLjreWai7kQlUjdj0OUtuRv1n9XCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57f828ee2270a71fcec46ccb948b6f6217a44cd3fb0d60f662db4f4ebca14f12","last_reissued_at":"2026-05-17T23:40:09.941961Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:09.941961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.08408","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZYpA7tTvQDuaYgelk4SjWL92eMHyjXLoC1KxF9GGQ0lFqFpuMG+l+h7iwp9DaFcjERMbfVqcxesYD0oeds+iCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T16:53:33.032261Z"},"content_sha256":"11aea0236e4cfc30a609e306718a244c5bc2d644421540ef0865d8e83f8b7ef4","schema_version":"1.0","event_id":"sha256:11aea0236e4cfc30a609e306718a244c5bc2d644421540ef0865d8e83f8b7ef4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:K74CR3RCOCTR7TWENTFZJC3PMI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Solutions of a class of nonlinear matrix equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Samik Pakhira, Sk Monowar Hossein, Snehasish Bose","submitted_at":"2019-07-19T08:43:39Z","abstract_excerpt":"In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \\geq 1$, $ A, B$ are nonsingular matrices and $Q$ is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08408","kind":"arxiv","version":1},"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":"2026-05-17T23:40:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"67/V1HTaX0NV8gjknRCX/bIqcUsK57EG1qF1bg8ox2/ekp4c5sa+iiWuWOepbPJ8XX4rYmMrY0LC6Gm0qJGXAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T16:53:33.033023Z"},"content_sha256":"c0f066f66f85e6961c4849edf7dd16269b54c6e29544047a83aa752e1761b337","schema_version":"1.0","event_id":"sha256:c0f066f66f85e6961c4849edf7dd16269b54c6e29544047a83aa752e1761b337"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K74CR3RCOCTR7TWENTFZJC3PMI/bundle.json","state_url":"https://pith.science/pith/K74CR3RCOCTR7TWENTFZJC3PMI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K74CR3RCOCTR7TWENTFZJC3PMI/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":"2026-06-11T16:53:33Z","links":{"resolver":"https://pith.science/pith/K74CR3RCOCTR7TWENTFZJC3PMI","bundle":"https://pith.science/pith/K74CR3RCOCTR7TWENTFZJC3PMI/bundle.json","state":"https://pith.science/pith/K74CR3RCOCTR7TWENTFZJC3PMI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K74CR3RCOCTR7TWENTFZJC3PMI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:K74CR3RCOCTR7TWENTFZJC3PMI","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":"f3bcde368ca8cfd2618623f2c8f5bc823a8325ddfc71594186351ded4814aa51","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2019-07-19T08:43:39Z","title_canon_sha256":"2f13edab83d3ac22fe28e6648e27c1cdc6e73527d51045623ba6e89396f86c5a"},"schema_version":"1.0","source":{"id":"1907.08408","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.08408","created_at":"2026-05-17T23:40:09Z"},{"alias_kind":"arxiv_version","alias_value":"1907.08408v1","created_at":"2026-05-17T23:40:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08408","created_at":"2026-05-17T23:40:09Z"},{"alias_kind":"pith_short_12","alias_value":"K74CR3RCOCTR","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"K74CR3RCOCTR7TWE","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"K74CR3RC","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:c0f066f66f85e6961c4849edf7dd16269b54c6e29544047a83aa752e1761b337","target":"graph","created_at":"2026-05-17T23:40:09Z","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":"In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \\geq 1$, $ A, B$ are nonsingular matrices and $Q$ is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation.","authors_text":"Samik Pakhira, Sk Monowar Hossein, Snehasish Bose","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2019-07-19T08:43:39Z","title":"Solutions of a class of nonlinear matrix equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08408","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":"sha256:11aea0236e4cfc30a609e306718a244c5bc2d644421540ef0865d8e83f8b7ef4","target":"record","created_at":"2026-05-17T23:40:09Z","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":"f3bcde368ca8cfd2618623f2c8f5bc823a8325ddfc71594186351ded4814aa51","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2019-07-19T08:43:39Z","title_canon_sha256":"2f13edab83d3ac22fe28e6648e27c1cdc6e73527d51045623ba6e89396f86c5a"},"schema_version":"1.0","source":{"id":"1907.08408","kind":"arxiv","version":1}},"canonical_sha256":"57f828ee2270a71fcec46ccb948b6f6217a44cd3fb0d60f662db4f4ebca14f12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"57f828ee2270a71fcec46ccb948b6f6217a44cd3fb0d60f662db4f4ebca14f12","first_computed_at":"2026-05-17T23:40:09.941961Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:09.941961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y26W8pWhQ9rAvN6numCqC+FWj8rBsk2uXER2s2dLe9BfGbewxmjtgrpdxHLjreWai7kQlUjdj0OUtuRv1n9XCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:09.942437Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.08408","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11aea0236e4cfc30a609e306718a244c5bc2d644421540ef0865d8e83f8b7ef4","sha256:c0f066f66f85e6961c4849edf7dd16269b54c6e29544047a83aa752e1761b337"],"state_sha256":"89a2bc2a6baece18dd3d2f2946eaf7189569ea37ce077d9172d81e3069756b6f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PfYxT2AGEfZ59XbtZ66ONuSQTmkK3LqmndtEIsgXaw29ji7h1b1S7x/DehFT9nJG5Y8yRmxYhl+m1XhCWPsfDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T16:53:33.037508Z","bundle_sha256":"23f86fd1f939525d9164f915f67d886ffa6e428424be3658aada73ffb844b010"}}