{"paper":{"title":"Cut norm discontinuity of triangular truncation of graphons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.CO","authors_text":"Teddy Mishura","submitted_at":"2021-10-13T18:54:49Z","abstract_excerpt":"The space of $L^p$ graphons, symmetric measurable functions $w: [0,1]^2 \\to \\mathbb{R}$ with finite $p$-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator $M_{\\chi}$ acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as $n\\to \\infty$, the norm of the triangular truncation operator $\\mathcal{T}_n$ on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of $L^p$ graphons, the norm growth of $\\mathcal{T}_n$ generalizes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.06984","kind":"arxiv","version":2},"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/2110.06984/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"}