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The intent of this paper will be to create a space $K$, pair of maps $g: C^{\\omega}(\\mathbb{R}) \\to K$ and $g': K \\to C^{\\omega}(\\mathbb{R}$), and operator $D^{k}: K \\to K$ such that the operator $D^{k}$ commutes with itself, the map $g$ embeds $C^{\\omega}(\\mathbb{R}$) isomorphically into $K$, and the following diagram commutes;\n  \\xymatrix{C^{\\omega}(\\mathbb{R}) \\ar[d]_{_{a}D_{x}^{k}} \\ar[r]^{g} & K \\ar[d]^{D^{k}}\n  C^{\\omega}(\\mathbb{R}) & K \\ar[l]^{g'}}\n  \\qquad This implies th"},"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":"1207.6610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-07-27T18:24:34Z","cross_cats_sorted":[],"title_canon_sha256":"d40211cc2d72204f412ab5abfc87b772653f4e4b466197a2faea335f49410271","abstract_canon_sha256":"8b003eb4ba73f17b328dfca642982b845868da602db09d3beb56095c3670c88e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:58.985697Z","signature_b64":"4DU4GsbRvzKHJQcAeKBOp9+qgr2hWOdeouwRHIxw9i8gW2NSNy89+OgeHMs2isPet0dOVANnj1L3XJmrH5lNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a571893eb46ac79e0bc6c7abd94fe7e07137a5d881504cefc4b5b623a27f6fc8","last_reissued_at":"2026-05-18T03:49:58.985012Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:58.985012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional Calculus - A Commutative Method on Real Analytic Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Matthew Parker","submitted_at":"2012-07-27T18:24:34Z","abstract_excerpt":"The traditional first approach to fractional calculus is via the Riemann-Liouville differintegral $_{a}D_{x}^{k}$. The intent of this paper will be to create a space $K$, pair of maps $g: C^{\\omega}(\\mathbb{R}) \\to K$ and $g': K \\to C^{\\omega}(\\mathbb{R}$), and operator $D^{k}: K \\to K$ such that the operator $D^{k}$ commutes with itself, the map $g$ embeds $C^{\\omega}(\\mathbb{R}$) isomorphically into $K$, and the following diagram commutes;\n  \\xymatrix{C^{\\omega}(\\mathbb{R}) \\ar[d]_{_{a}D_{x}^{k}} \\ar[r]^{g} & K \\ar[d]^{D^{k}}\n  C^{\\omega}(\\mathbb{R}) & K \\ar[l]^{g'}}\n  \\qquad This implies th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6610","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":"1207.6610","created_at":"2026-05-18T03:49:58.985113+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.6610v1","created_at":"2026-05-18T03:49:58.985113+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6610","created_at":"2026-05-18T03:49:58.985113+00:00"},{"alias_kind":"pith_short_12","alias_value":"UVYYSPVUNLDZ","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"UVYYSPVUNLDZ4C6G","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"UVYYSPVU","created_at":"2026-05-18T12:27:25.539911+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/UVYYSPVUNLDZ4C6GY6V5ST7H4B","json":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B.json","graph_json":"https://pith.science/api/pith-number/UVYYSPVUNLDZ4C6GY6V5ST7H4B/graph.json","events_json":"https://pith.science/api/pith-number/UVYYSPVUNLDZ4C6GY6V5ST7H4B/events.json","paper":"https://pith.science/paper/UVYYSPVU"},"agent_actions":{"view_html":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B","download_json":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B.json","view_paper":"https://pith.science/paper/UVYYSPVU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.6610&json=true","fetch_graph":"https://pith.science/api/pith-number/UVYYSPVUNLDZ4C6GY6V5ST7H4B/graph.json","fetch_events":"https://pith.science/api/pith-number/UVYYSPVUNLDZ4C6GY6V5ST7H4B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B/action/storage_attestation","attest_author":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B/action/author_attestation","sign_citation":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B/action/citation_signature","submit_replication":"https://pith.science/pith/UVYYSPVUNLDZ4C6GY6V5ST7H4B/action/replication_record"}},"created_at":"2026-05-18T03:49:58.985113+00:00","updated_at":"2026-05-18T03:49:58.985113+00:00"}