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تسجيل مستخدم جديدOn the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $Bbb R^D$
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Let $Dgeq 2$, $Ssubset mathbb R^D$ be finite and let $phi:Sto mathbb R^D$ with $phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $phi$ can be extended to a function $Phi:mathbb R^Dto mathbb R^D$ which is a smooth small distortion on $mathbb R^D$. Our main results are in addition to Whitney extensions, results on interpolation and alignment of data in $mathbb R^D$ and complement those of [14,15,20].
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In this paper, we study the following problem: Let $Dgeq 2$ and let $Esubset mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $phi:Eto mathbb R^D$ with $phi$ a small distortion on $E$. How can one decide whether $p
hi$ extends to a smooth small distortion $Phi:mathbb R^Dto mathbb R^D$ which agrees with $phi$ on $E$. We also ask how to decide if in addition $Phi$ can be approximated well by certain rigid and non-rigid motions from $mathbb R^Dto mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $mathbb R^D$.
In this memoir, we develop a general framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in $mathbb R^D$ and the Whitney near isometry extension problem for discrete and non-discrete subsets of $mathb
b R^D$ with certain geometries. In addition, we survey related work of ours on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems in harmonic analysis, computer vision, manifold learning and signal processing connected to our work are given. A significant portion of the work in this memoir is based on joint research with Charles Fefferman in the papers [48], [49], [50], [51].
Let $ f $ be a real-valued function on a compact subset in $ mathbb{R}^n $. We show how to decide if $ f $ extends to a nonnegative and $ C^1 $ function on $ mathbb{R}^n $. There has been no known result for nonnegative $ C^m $ extension from a gener
al compact set $ E $ when $ m > 0 $. The nonnegative extension problem for $ m geq 2 $ remains open.
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes. If in any o
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