Implicit Functions, Lipschitz Maps, and Stability in Optimization

1994 ◽  
Vol 19 (3) ◽  
pp. 753-768 ◽  
Author(s):  
Asen L. Dontchev ◽  
William W. Hager
Keyword(s):  
Implicit Functions ◽  
Lipschitz Maps

scholarly journals On the gradient set of Lipschitz maps

2008 ◽  
Vol 2008 (625) ◽  
Author(s):  
Bernd Kirchheim ◽  
László Székelyhidi
Keyword(s):  
Lipschitz Maps

Implicit functions lab

2005 ◽  
Vol 37 (3) ◽  
pp. 376-376
Author(s):  
Edmond C. Prakash
Keyword(s):  
Implicit Functions

Interactive 3D medical image segmentation with energy-minimizing implicit functions

2011 ◽  
Vol 35 (2) ◽  
pp. 275-287 ◽  
Author(s):  
Frank Heckel ◽  
Olaf Konrad ◽  
Horst Karl Hahn ◽  
Heinz-Otto Peitgen
Keyword(s):  
Image Segmentation ◽  
Medical Image ◽  
Medical Image Segmentation ◽  
Implicit Functions ◽  
Interactive 3D

Implicit Function Alteration via Radius Mapping and Direct Function Modification

1995 ◽  
Author(s):  
Mark T. Ensz ◽  
Mark A. Ganter ◽  
Duane W. Storti
Keyword(s):  
Generalized Function ◽  
Implicit Function ◽  
Implicit Functions ◽  
Direct Function ◽  
Surface Features ◽  
And Function ◽  
Implicit Solid

Abstract In this paper, we present two closely related techniques, radius mapping, and direct function modification, that allow for the alteration of both the major geometry and surface features of implicit solids. The first technique, radius mapping, is applied to implicit functions generated by swept solid techniques. A more generalized technique, direct function modification, allows for the mapping of generalized function modifiers onto any implicit solid. Both the radius functions and the function modifiers can be either algebraic or non-algebraic in nature. Techniques for generating both algebraic and non-algebraic radius functions and function modifiers are given along with several examples of their use on both swept solids and implicit functions.

scholarly journals A DEFINABLE -ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

2015 ◽  
Vol 17 (1) ◽  
pp. 39-57 ◽  
Author(s):  
Raf Cluckers ◽  
Florent Martin
Keyword(s):  
Direct Application ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions ◽  
Topological Closure

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

Differentiation of Implicit Functions

Calculus I ◽  
1975 ◽  
pp. 44-49
Author(s):  
Brian Knight ◽  
Roger Adams
Keyword(s):  
Implicit Functions
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