{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:XEP4KJCOL67ZXPV6AAGHJPFPI7","short_pith_number":"pith:XEP4KJCO","schema_version":"1.0","canonical_sha256":"b91fc5244e5fbf9bbebe000c74bcaf47c664e080f2cc4b232a8727a471a83962","source":{"kind":"arxiv","id":"0908.3307","version":1},"attestation_state":"computed","paper":{"title":"The Gateaux Derivative of Map over Division Ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Aleks Kleyn","submitted_at":"2009-08-23T15:18:40Z","abstract_excerpt":"I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent.\n  I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0908.3307","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2009-08-23T15:18:40Z","cross_cats_sorted":[],"title_canon_sha256":"a7885eb9b4feb5d6588bd41394ca4182701408e3da6b14fd166e2e100680a172","abstract_canon_sha256":"46448cf60b6e44b7306eb057b4e99f24a12a79bb2e4c55f193f8b657ebb47edf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:11.206594Z","signature_b64":"yviPsd+diAJcOG8/vsfcmEBg8e+uZ/eppM+fQ1Nji53moTUAD3ipZDPY/1DkBKzszz/v6mzmd/cNDbbXnDt+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b91fc5244e5fbf9bbebe000c74bcaf47c664e080f2cc4b232a8727a471a83962","last_reissued_at":"2026-05-18T03:50:11.205831Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:11.205831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Gateaux Derivative of Map over Division Ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Aleks Kleyn","submitted_at":"2009-08-23T15:18:40Z","abstract_excerpt":"I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent.\n  I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3307","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0908.3307","created_at":"2026-05-18T03:50:11.205960+00:00"},{"alias_kind":"arxiv_version","alias_value":"0908.3307v1","created_at":"2026-05-18T03:50:11.205960+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.3307","created_at":"2026-05-18T03:50:11.205960+00:00"},{"alias_kind":"pith_short_12","alias_value":"XEP4KJCOL67Z","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"XEP4KJCOL67ZXPV6","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"XEP4KJCO","created_at":"2026-05-18T12:26:02.257875+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7","json":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7.json","graph_json":"https://pith.science/api/pith-number/XEP4KJCOL67ZXPV6AAGHJPFPI7/graph.json","events_json":"https://pith.science/api/pith-number/XEP4KJCOL67ZXPV6AAGHJPFPI7/events.json","paper":"https://pith.science/paper/XEP4KJCO"},"agent_actions":{"view_html":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7","download_json":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7.json","view_paper":"https://pith.science/paper/XEP4KJCO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0908.3307&json=true","fetch_graph":"https://pith.science/api/pith-number/XEP4KJCOL67ZXPV6AAGHJPFPI7/graph.json","fetch_events":"https://pith.science/api/pith-number/XEP4KJCOL67ZXPV6AAGHJPFPI7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/action/storage_attestation","attest_author":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/action/author_attestation","sign_citation":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/action/citation_signature","submit_replication":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/action/replication_record"}},"created_at":"2026-05-18T03:50:11.205960+00:00","updated_at":"2026-05-18T03:50:11.205960+00:00"}