Relevant bibliographies by topics / Kirszbraun

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  2. Dissertations / Theses

Academic literature on the topic 'Kirszbraun'

Author: Grafiati
Published: 4 June 2021
Last updated: 2 February 2022

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Journal articles on the topic "Kirszbraun"

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Chandgotia, Nishant, Igor Pak, and Martin Tassy. "Kirszbraun-type theorems for graphs." Journal of Combinatorial Theory, Series B 137 (July 2019): 10–24. http://dx.doi.org/10.1016/j.jctb.2018.11.007.

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Aschenbrenner, Matthias, and Andreas Fischer. "Definable versions of theorems by Kirszbraun and Helly." Proceedings of the London Mathematical Society 102, no. 3 (2010): 468–502. http://dx.doi.org/10.1112/plms/pdq029.

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Reich, Simeon, and Stephen Simons. "Fenchel duality, Fitzpatrick functions and the Kirszbraun–Valentine extension theorem." Proceedings of the American Mathematical Society 133, no. 9 (2005): 2657–60. http://dx.doi.org/10.1090/s0002-9939-05-07983-9.

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Kopecká, Eva. "Bootstrapping Kirszbraun's extension theorem." Fundamenta Mathematicae 217, no. 1 (2012): 13–19. http://dx.doi.org/10.4064/fm217-1-2.

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Cluckers, Raf, and Florent Martin. "A DEFINABLE -ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS." Journal of the Institute of Mathematics of Jussieu 17, no. 1 (2015): 39–57. http://dx.doi.org/10.1017/s1474748015000390.

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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.
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Akopyan, A. V., and A. S. Tarasov. "A constructive proof of Kirszbraun’s theorem." Mathematical Notes 84, no. 5-6 (2008): 725–28. http://dx.doi.org/10.1134/s000143460811014x.

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7

Lang, U., and V. Schroeder. "Kirszbraun's Theorem and Metric Spaces of Bounded Curvature." Geometric And Functional Analysis 7, no. 3 (1997): 535–60. http://dx.doi.org/10.1007/s000390050018.

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Logaritsch, Philippe, and Andrea Marchese. "Kirszbraun’s extension theorem fails for Almgren’s multiple valued functions." Archiv der Mathematik 102, no. 5 (2014): 455–58. http://dx.doi.org/10.1007/s00013-014-0642-4.

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Krieger, Andrew, Georg Menz, and Martin Tassy. "Deducing a Variational Principle with Minimal A Priori Assumptions." Electronic Journal of Combinatorics 27, no. 4 (2020). http://dx.doi.org/10.37236/9121.

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We study the well-known variational and large deviation principle for graph homomorphisms from $\mathbb{Z}^m$ to $\mathbb{Z}$. We provide a robust method to deduce those principles under minimal a priori assumptions. The only ingredient specific to the model is a discrete Kirszbraun theorem i.e. an extension theorem for graph homomorphisms. All other ingredients are of a general nature not specific to the model. They include elementary combinatorics, the compactness of Lipschitz functions, and a simplicial Rademacher theorem. Compared to the literature, our proof does not need any other preliminary results like e.g. concentration or strict convexity of the local surface tension. Therefore, the method is very robust and extends to more complex and subtle models, as e.g. the homogenization of limit shapes or graph-homomorphisms to a regular tree.
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Azagra, Daniel, Erwan Le Gruyer, and Carlos Mudarra. "Kirszbraun’s Theorem via an Explicit Formula." Canadian Mathematical Bulletin, April 29, 2020, 1–12. http://dx.doi.org/10.4153/s0008439520000314.

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Abstract Let $X,Y$ be two Hilbert spaces, let E be a subset of $X,$ and let $G\colon E \to Y$ be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists $\tilde {G} : X \to Y$ with $\tilde {G}=G$ on E and $ \operatorname {\mathrm {Lip}}(\tilde {G})= \operatorname {\mathrm {Lip}}(G).$ In this note we show that in fact the function $\tilde {G}:=\nabla _Y( \operatorname {\mathrm {conv}} (g))( \cdot , 0)$ , where $$\begin{align*}g(x,y) = \inf_{z \in E} \Big\lbrace \langle G(z), y \rangle + \frac{\operatorname{\mathrm{Lip}}(G)}{2} \|(x-z,y)\|^2 \Big\rbrace + \frac{\operatorname{\mathrm{Lip}}(G)}{2}\|(x,y)\|^2, \end{align*}$$ defines such an extension. We apply this formula to get an extension result for strongly biLipschitz mappings. Related to the latter, we also consider extensions of $C^{1,1}$ strongly convex functions.
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Dissertations / Theses on the topic "Kirszbraun"

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Phan, Thanh Viet. "Extensions lipschitziennes minimales." Thesis, Rennes, INSA, 2015. http://www.theses.fr/2015ISAR0027/document.

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Cette thèse est consacrée aux quelques problèmes mathématiques concernant les extensions minimales de Lipschitz. Elle est organisée de manière suivante. Le chapitre 1 est dédié à l’introduction des extensions minimales de Lipschitz. Dans le chapitre 2, nous étudions la relation entre la constante de Lipschitz d’ 1-field et la constante de Lipschitz du gradient associée à ce 1-field. Nous proposons deux formules explicites Sup-Inf, qui sont des extensions extrêmes minimales de Lipschitz d’1-field. Nous expliquons comment les utiliser pour construire les extensions minimales de Lipschitz pour les applications Rmà Rn . Par ailleurs, nous montrons que les extensions de Wells d’1- fields sont les extensions absolument minimales de Lipschitz (AMLE) lorsque le domaine d’expansion d’1-field est infini. Un contreexemple est présenté afin de montrer que ce résultat n’est pas vrai en général. Dans le chapitre 3, nous étudions la version discrète de l’existence et l’unicité de l’AMLE. Nous montrons que la fonction tight introduite par Sheffield and Smart est l’extension de Kirszbraun. Dans le cas réel, nous pouvons montrer que cette extension est unique. De plus, nous proposons un algorithme qui permet de calculer efficacement la valeur de l’extension de Kirszbraun en complexité polynomiale. Pour conclure, nous décrivons quelques pistes pour la future recherche, qui sont liées au sujet présenté dans ce manuscrit<br>The thesis is concerned to some mathematical problems on minimal Lipschitz extensions. Chapter 1: We introduce some basic background about minimal Lipschitz extension (MLE) problems. Chapter 2: We study the relationship between the Lipschitz constant of 1-field and the Lipschitz constant of the gradient associated with this 1-field. We produce two Sup-Inf explicit formulas which are two extremal minimal Lipschitz extensions for 1-fields. We explain how to use the Sup-Inf explicit minimal Lipschitz extensions for 1-fields to construct minimal Lipschitz extension of mappings from Rm to Rn. Moreover, we show that Wells’s extensions of 1-fields are absolutely minimal Lipschitz extensions (AMLE) when the domain of 1-field to expand is finite. We provide a counter-example showing that this result is false in general. Chapter 3: We study the discrete version of the existence and uniqueness of AMLE. We prove that the tight function introduced by Sheffield and Smart is a Kirszbraun extension. In the realvalued case, we prove that the Kirszbraun extension is unique. Moreover, we produce a simple algorithm which calculates efficiently the value of the Kirszbraun extension in polynomial time. Chapter 4: We describe some problems for future research, which are related to the subject represented in the thesis
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