{"paper":{"title":"Geometry of Curves in $\\mathbb R^n$, Singular Value Decomposition, and Hankel Determinants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bruce Draper, Chris Peterson, Michael Kirby, Robert Arn, Xavier \\'Alvarez-Vizoso","submitted_at":"2015-11-16T15:58:05Z","abstract_excerpt":"Let $\\gamma: I \\rightarrow \\mathbb R^n$ be a parametric curve of class $C^{n+1}$, regular of order $n$. The Frenet-Serret apparatus of $\\gamma$ at $\\gamma(t)$ consists of a frame $e_1(t), \\dots , e_n(t)$ and generalized curvature values $\\kappa_1(t), \\dots, \\kappa_{n-1}(t)$. Associated with each point of $\\gamma$ there are also local singular vectors $u_1(t), \\dots, u_n(t)$ and local singular values $\\sigma_1(t), \\dots, \\sigma_{n}(t)$. This local information is obtained by considering a limit, as $\\epsilon$ goes to zero, of covariance matrices defined along $\\gamma$ within an $\\epsilon$-ball c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05008","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":""},"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"}