{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:NJFAJUYAJDZ3RT5GDKH2BA36QT","short_pith_number":"pith:NJFAJUYA","canonical_record":{"source":{"id":"1701.08150","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-01-27T18:54:54Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"fe73a50f7570af79ee1f1f737a725f4cce7c4dc4118220ae0afaab1b8d3d9ab8","abstract_canon_sha256":"6825054c850c101c2c303f21f26acf9e2c5e70cb7d8e94b859d9f4b6d38810b1"},"schema_version":"1.0"},"canonical_sha256":"6a4a04d30048f3b8cfa61a8fa0837e84efe665f628e418a46412ff9124836070","source":{"kind":"arxiv","id":"1701.08150","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.08150","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"1701.08150v1","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.08150","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"NJFAJUYAJDZ3","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"NJFAJUYAJDZ3RT5G","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"NJFAJUYA","created_at":"2026-05-18T12:31:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:NJFAJUYAJDZ3RT5GDKH2BA36QT","target":"record","payload":{"canonical_record":{"source":{"id":"1701.08150","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-01-27T18:54:54Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"fe73a50f7570af79ee1f1f737a725f4cce7c4dc4118220ae0afaab1b8d3d9ab8","abstract_canon_sha256":"6825054c850c101c2c303f21f26acf9e2c5e70cb7d8e94b859d9f4b6d38810b1"},"schema_version":"1.0"},"canonical_sha256":"6a4a04d30048f3b8cfa61a8fa0837e84efe665f628e418a46412ff9124836070","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:58.731058Z","signature_b64":"2t8Fv4DSfHpfrsMCtSTM/CAdXOwaRfDfPJMq9ComLAtoaozNlbm3rAkYh51s+XCif5a8wAzamFB6J/qKJ9lvDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a4a04d30048f3b8cfa61a8fa0837e84efe665f628e418a46412ff9124836070","last_reissued_at":"2026-05-18T00:51:58.730478Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:58.730478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.08150","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":"2026-05-18T00:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ENmcgoGPrX4uAEwfVtAVJ3iXv03uUH09chn4QJ5/8qy4eSpaTrm593P8vAsIH8xcRDkGeEcx8qZgdWJmkEZtDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T18:40:23.073559Z"},"content_sha256":"a23b5bdf5cdd16604eea706c67a6e0b9bd3571f89c7979f8de811f50829bb622","schema_version":"1.0","event_id":"sha256:a23b5bdf5cdd16604eea706c67a6e0b9bd3571f89c7979f8de811f50829bb622"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:NJFAJUYAJDZ3RT5GDKH2BA36QT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Grioli's Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Andreas Fischle, Patrizio Neff","submitted_at":"2017-01-27T18:54:54Z","abstract_excerpt":"Let $F \\in {\\rm GL}^+(3)$ and consider the right polar decomposition $F = R_p(F)\\cdot U$ into an orthogonal factor $R_p(F) \\in {\\rm SO}(3)$ and a symmetric, positive definite factor $U(F) = \\sqrt{F^TF} \\in {\\rm Psym}(3)$. In 1940 Giuseppe Grioli proved that $$ {\\rm argmin}_{R \\in {\\rm SO}(3)} ||R^TF - 1{||}^2 \\quad=\\quad \\{\\,R_p(F)\\,\\} \\quad=\\quad {\\rm argmin}_{R \\in {\\rm SO}(3)} ||F - R{||}^2\\;. $$ This variational characterization of the orthogonal factor $R_p(F) \\in {\\rm SO}(n)$ holds in any dimension $n \\geq 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we character"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08150","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":"2026-05-18T00:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m7lzHBQH7dqGMTvP4Zk4nD80ve2XLH2x++w8OYB3lKsv4GAAYgHLYtM7BcZuICL1RJ4sGhwcV/HVfM0uJvWJDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T18:40:23.073904Z"},"content_sha256":"75c7d78eeef74e5f2b730439f3c33df7e9467d1099a5f02980502539d7332e57","schema_version":"1.0","event_id":"sha256:75c7d78eeef74e5f2b730439f3c33df7e9467d1099a5f02980502539d7332e57"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT/bundle.json","state_url":"https://pith.science/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT/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":"2026-06-01T18:40:23Z","links":{"resolver":"https://pith.science/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT","bundle":"https://pith.science/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT/bundle.json","state":"https://pith.science/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NJFAJUYAJDZ3RT5GDKH2BA36QT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:NJFAJUYAJDZ3RT5GDKH2BA36QT","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":"6825054c850c101c2c303f21f26acf9e2c5e70cb7d8e94b859d9f4b6d38810b1","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-01-27T18:54:54Z","title_canon_sha256":"fe73a50f7570af79ee1f1f737a725f4cce7c4dc4118220ae0afaab1b8d3d9ab8"},"schema_version":"1.0","source":{"id":"1701.08150","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.08150","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"1701.08150v1","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.08150","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"NJFAJUYAJDZ3","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"NJFAJUYAJDZ3RT5G","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"NJFAJUYA","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:75c7d78eeef74e5f2b730439f3c33df7e9467d1099a5f02980502539d7332e57","target":"graph","created_at":"2026-05-18T00:51:58Z","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":"Let $F \\in {\\rm GL}^+(3)$ and consider the right polar decomposition $F = R_p(F)\\cdot U$ into an orthogonal factor $R_p(F) \\in {\\rm SO}(3)$ and a symmetric, positive definite factor $U(F) = \\sqrt{F^TF} \\in {\\rm Psym}(3)$. In 1940 Giuseppe Grioli proved that $$ {\\rm argmin}_{R \\in {\\rm SO}(3)} ||R^TF - 1{||}^2 \\quad=\\quad \\{\\,R_p(F)\\,\\} \\quad=\\quad {\\rm argmin}_{R \\in {\\rm SO}(3)} ||F - R{||}^2\\;. $$ This variational characterization of the orthogonal factor $R_p(F) \\in {\\rm SO}(n)$ holds in any dimension $n \\geq 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we character","authors_text":"Andreas Fischle, Patrizio Neff","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-01-27T18:54:54Z","title":"Grioli's Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08150","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:a23b5bdf5cdd16604eea706c67a6e0b9bd3571f89c7979f8de811f50829bb622","target":"record","created_at":"2026-05-18T00:51:58Z","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":"6825054c850c101c2c303f21f26acf9e2c5e70cb7d8e94b859d9f4b6d38810b1","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-01-27T18:54:54Z","title_canon_sha256":"fe73a50f7570af79ee1f1f737a725f4cce7c4dc4118220ae0afaab1b8d3d9ab8"},"schema_version":"1.0","source":{"id":"1701.08150","kind":"arxiv","version":1}},"canonical_sha256":"6a4a04d30048f3b8cfa61a8fa0837e84efe665f628e418a46412ff9124836070","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a4a04d30048f3b8cfa61a8fa0837e84efe665f628e418a46412ff9124836070","first_computed_at":"2026-05-18T00:51:58.730478Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:58.730478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2t8Fv4DSfHpfrsMCtSTM/CAdXOwaRfDfPJMq9ComLAtoaozNlbm3rAkYh51s+XCif5a8wAzamFB6J/qKJ9lvDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:58.731058Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.08150","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a23b5bdf5cdd16604eea706c67a6e0b9bd3571f89c7979f8de811f50829bb622","sha256:75c7d78eeef74e5f2b730439f3c33df7e9467d1099a5f02980502539d7332e57"],"state_sha256":"7abe904b14c6aff159f326252de8fd195b28bb5ece3e6d3ca541eb12cb995fcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xWHCEHtP0etnExPpD1JYvokIPghtcN8CRu/4MCb60pdLZ4EJGcQnkpmFxsYVJPtjtoAHvJvu9RRaacWGxIkABg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T18:40:23.075829Z","bundle_sha256":"716870ce027df3f077df785e20d93ee7059c4fe6dd9ec9fc8a4d36ad3097fba3"}}