{"paper":{"title":"Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.CA","authors_text":"Hanpeng Gao, Shengda Liu, Yucheng Wang, Yu-Zhe Liu","submitted_at":"2024-05-31T19:47:45Z","abstract_excerpt":"We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\\mathbb{I}_1 \\subseteq {\\mathit{\\Lambda}}_1$ and $\\mathbb{I}_2 \\subseteq {\\mathit{\\Lambda}}_2$ of two finite-dimensional algebras, under the influence of a mapping $\\omega$, which can be an injection or a bijection. We explore four specific cases:\n  $\\bullet$ $\\omega$ as a monotone non-decreasing and right-continuous function;\n  $\\bullet$ $\\omega$ as an injective, absolutely continuous function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.00161","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.00161/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}