{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:ZOW7V2XRCM6GUSBJLYLHIE3SVF","short_pith_number":"pith:ZOW7V2XR","schema_version":"1.0","canonical_sha256":"cbadfaeaf1133c6a48295e16741372a95fb71c0be0f4ed0ab276efcdfbfe5010","source":{"kind":"arxiv","id":"1804.09227","version":1},"attestation_state":"computed","paper":{"title":"Fractional powers of vector operators and fractional Fourier's law in a Hilbert space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Fabrizio Colombo, Jonathan Gantner","submitted_at":"2018-04-24T19:41:41Z","abstract_excerpt":"In this paper we give a concrete application of the spectral theory based on the notion of $S$-spectrum to fractional diffusion process. Precisely, we consider the Fourier law for the propagation of the heat in non homogeneous materials, that is the heat flow is given by the vector operator: $$ T=e_1\\,a(x)\\partial_{x_1} + e_2\\,b(x)\\partial_{x_2} + e_3\\,c(x)\\partial_{x_3} $$ where $e_\\ell$, $\\ell=1,2,3$ are orthogonal unit vectors in $\\mathbb{R}^3$, $a$, $b$, $c$ are given real valued functions that depend on the space variables $x=(x_1,x_2,x_3)$, and possibly also on time.\n  Using the $H^\\inft"},"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":"1804.09227","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-24T19:41:41Z","cross_cats_sorted":[],"title_canon_sha256":"b68acd36efe4ffd8e8b8a1d16c726ad69dec95389ec46aa6b4b5742f01ce5815","abstract_canon_sha256":"31dc4dd0fa57aacae4cb14054c8d520cf7af2b48fd7f710ccd4fa2f335fa8776"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:14.420736Z","signature_b64":"a1QO2qrOJLtgMhxNe+7fBuq2PJNckSuiomdT9jeTBDxt/i15v9z0Rx6I8kiDT9zqpAoTcdV0ZF2P+wib6PpqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbadfaeaf1133c6a48295e16741372a95fb71c0be0f4ed0ab276efcdfbfe5010","last_reissued_at":"2026-05-18T00:09:14.420077Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:14.420077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional powers of vector operators and fractional Fourier's law in a Hilbert space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Fabrizio Colombo, Jonathan Gantner","submitted_at":"2018-04-24T19:41:41Z","abstract_excerpt":"In this paper we give a concrete application of the spectral theory based on the notion of $S$-spectrum to fractional diffusion process. Precisely, we consider the Fourier law for the propagation of the heat in non homogeneous materials, that is the heat flow is given by the vector operator: $$ T=e_1\\,a(x)\\partial_{x_1} + e_2\\,b(x)\\partial_{x_2} + e_3\\,c(x)\\partial_{x_3} $$ where $e_\\ell$, $\\ell=1,2,3$ are orthogonal unit vectors in $\\mathbb{R}^3$, $a$, $b$, $c$ are given real valued functions that depend on the space variables $x=(x_1,x_2,x_3)$, and possibly also on time.\n  Using the $H^\\inft"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09227","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":"1804.09227","created_at":"2026-05-18T00:09:14.420189+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.09227v1","created_at":"2026-05-18T00:09:14.420189+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.09227","created_at":"2026-05-18T00:09:14.420189+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZOW7V2XRCM6G","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZOW7V2XRCM6GUSBJ","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZOW7V2XR","created_at":"2026-05-18T12:33:07.085635+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/ZOW7V2XRCM6GUSBJLYLHIE3SVF","json":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF.json","graph_json":"https://pith.science/api/pith-number/ZOW7V2XRCM6GUSBJLYLHIE3SVF/graph.json","events_json":"https://pith.science/api/pith-number/ZOW7V2XRCM6GUSBJLYLHIE3SVF/events.json","paper":"https://pith.science/paper/ZOW7V2XR"},"agent_actions":{"view_html":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF","download_json":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF.json","view_paper":"https://pith.science/paper/ZOW7V2XR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.09227&json=true","fetch_graph":"https://pith.science/api/pith-number/ZOW7V2XRCM6GUSBJLYLHIE3SVF/graph.json","fetch_events":"https://pith.science/api/pith-number/ZOW7V2XRCM6GUSBJLYLHIE3SVF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF/action/storage_attestation","attest_author":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF/action/author_attestation","sign_citation":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF/action/citation_signature","submit_replication":"https://pith.science/pith/ZOW7V2XRCM6GUSBJLYLHIE3SVF/action/replication_record"}},"created_at":"2026-05-18T00:09:14.420189+00:00","updated_at":"2026-05-18T00:09:14.420189+00:00"}