{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:XEP4KJCOL67ZXPV6AAGHJPFPI7","short_pith_number":"pith:XEP4KJCO","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"},"canonical_sha256":"b91fc5244e5fbf9bbebe000c74bcaf47c664e080f2cc4b232a8727a471a83962","source":{"kind":"arxiv","id":"0908.3307","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.3307","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"arxiv_version","alias_value":"0908.3307v1","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.3307","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"pith_short_12","alias_value":"XEP4KJCOL67Z","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"XEP4KJCOL67ZXPV6","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"XEP4KJCO","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:XEP4KJCOL67ZXPV6AAGHJPFPI7","target":"record","payload":{"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"},"canonical_sha256":"b91fc5244e5fbf9bbebe000c74bcaf47c664e080f2cc4b232a8727a471a83962","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"},"source_kind":"arxiv","source_id":"0908.3307","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":"2026-05-18T03:50:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gZEztz33+Yd0hCwtSPDdwwekZxbAkCJYlkqecPdrWfLosArg2bdv+izo+sQ3gvdc8myhoSrtteOCQdPGrQXdDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:36:27.996026Z"},"content_sha256":"b2c6af16738706850baad1676060a3ead97a2f742c8cce00a7d8719b02117f3a","schema_version":"1.0","event_id":"sha256:b2c6af16738706850baad1676060a3ead97a2f742c8cce00a7d8719b02117f3a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:XEP4KJCOL67ZXPV6AAGHJPFPI7","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"smIctPbgBj9IqJlISvOeUVkJ8kiEjb7E0KRpa/QDmUNlsB4AFkkm6YAhM9hx9xh60JlFLBIk/UQ5ugWKKo6IDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:36:27.996413Z"},"content_sha256":"131af12a2476839f538f0781bab83b05b3e32e719c856cbeefc425c153f1ba71","schema_version":"1.0","event_id":"sha256:131af12a2476839f538f0781bab83b05b3e32e719c856cbeefc425c153f1ba71"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/bundle.json","state_url":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/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":"2026-06-09T16:36:27Z","links":{"resolver":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7","bundle":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/bundle.json","state":"https://pith.science/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XEP4KJCOL67ZXPV6AAGHJPFPI7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:XEP4KJCOL67ZXPV6AAGHJPFPI7","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":"46448cf60b6e44b7306eb057b4e99f24a12a79bb2e4c55f193f8b657ebb47edf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2009-08-23T15:18:40Z","title_canon_sha256":"a7885eb9b4feb5d6588bd41394ca4182701408e3da6b14fd166e2e100680a172"},"schema_version":"1.0","source":{"id":"0908.3307","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.3307","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"arxiv_version","alias_value":"0908.3307v1","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.3307","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"pith_short_12","alias_value":"XEP4KJCOL67Z","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"XEP4KJCOL67ZXPV6","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"XEP4KJCO","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:131af12a2476839f538f0781bab83b05b3e32e719c856cbeefc425c153f1ba71","target":"graph","created_at":"2026-05-18T03:50:11Z","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":"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","authors_text":"Aleks Kleyn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2009-08-23T15:18:40Z","title":"The Gateaux Derivative of Map over Division Ring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3307","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:b2c6af16738706850baad1676060a3ead97a2f742c8cce00a7d8719b02117f3a","target":"record","created_at":"2026-05-18T03:50:11Z","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":"46448cf60b6e44b7306eb057b4e99f24a12a79bb2e4c55f193f8b657ebb47edf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2009-08-23T15:18:40Z","title_canon_sha256":"a7885eb9b4feb5d6588bd41394ca4182701408e3da6b14fd166e2e100680a172"},"schema_version":"1.0","source":{"id":"0908.3307","kind":"arxiv","version":1}},"canonical_sha256":"b91fc5244e5fbf9bbebe000c74bcaf47c664e080f2cc4b232a8727a471a83962","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b91fc5244e5fbf9bbebe000c74bcaf47c664e080f2cc4b232a8727a471a83962","first_computed_at":"2026-05-18T03:50:11.205831Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:11.205831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yviPsd+diAJcOG8/vsfcmEBg8e+uZ/eppM+fQ1Nji53moTUAD3ipZDPY/1DkBKzszz/v6mzmd/cNDbbXnDt+AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:11.206594Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.3307","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2c6af16738706850baad1676060a3ead97a2f742c8cce00a7d8719b02117f3a","sha256:131af12a2476839f538f0781bab83b05b3e32e719c856cbeefc425c153f1ba71"],"state_sha256":"95cb6117b5029a0246ea02a40245ebaf2240bea1028a86d68f26bff9efce504e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WChh/uliTVd+FF77gweAtTTB750pswH9eeevje63utLJROk4rAlvCjxijyn2TgJORE3DCHvN/xx2OHM0OHsDDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T16:36:27.998497Z","bundle_sha256":"bc0fbf891bd868535b9ef0697a8502c392b2227604befc88d08ba1f09255c415"}}