Well-posedness of optimization problems and Hausdorff metric on partial maps

Alessandro Caterino; Rita Ceppitelli; Ľubica Holà

Displaying similar documents to “Well-posedness of optimization problems and Hausdorff metric on partial maps”

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

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Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

A Note on $\mathcal{R}$ -Maps

Zbigniew Duszyński (2007)

Bollettino dell'Unione Matematica Italiana

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Richness of the class of $\mathcal{R}$ -maps [3] is investigated. Several consequences for the theory of quasi- $\mathcal{H}$ -closed spaces are indicated.

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0, σ}$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0, σ}$ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS $ϕ_{σ, α}^{n, k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

Extending generalized Whitney maps

Ivan Lončar (2017)

Archivum Mathematicum

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For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

On the Hausdorff Dimension of Topological Subspaces

Tomasz Szarek, Maciej Ślęczka (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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It is shown that every Polish space X with $d i m_{T} X \geq d$ admits a compact subspace Y such that $d i m_{H} Y \geq d$ where $d i m_{T}$ and $d i m_{H}$ denote the topological and Hausdorff dimensions, respectively.

Adjacent dyadic systems and the $L^{p}$ -boundedness of shift operators in metric spaces revisited

Olli Tapiola (2016)

Colloquium Mathematicae

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With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^{p}$ -boundedness of shift operators acting on functions $f \in L^{p} (X; E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

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We give a meaning to derivative of a function $u ℝ \to X$ , where $X$ is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space $𝒯_{x} X$ of $x \in X$ . Let $u, v [0, 1) \to X$ , $u (0) = v (0)$ be continuous at zero. Then by the definition $u$ and $v$ are in the same equivalence class if they are tangent at zero, that is if $lim_{h \to 0^{+}} \frac{d (u (h), v (h))}{h} = 0 .$ By...

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

The equation $\bar{\partial}u=f$ the intersection of pseudoconvex domains

Alessandro Perotti (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Viene studiata l'equazione $\bar{\partial}u=f$ per le forme regolari sulla chiusura dell'intersezione di $k$ domini pseudoconvessi. Si costruisce un operatore soluzione in forma integrale e sotto ipotesi opportune si ottengono stime della soluzione nelle norme $\mathbf{C}^{k}$ .

Some Separation Axioms Via Ideals

D. Sivaraj, V. Renuka Devi (2007)

Bollettino dell'Unione Matematica Italiana

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We introduce a new class of spaces, called Hausdorff modulo $\mathcal{I}$ or $T_{2}$ mod $\mathcal{I}$ spaces with respect to an ideal $\mathcal{I}$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $\mathcal{I}$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $\mathcal{I}$ -compact subsets in Hausdorff modulo $\mathcal{I}$ spaces and generalize some results of Hamlett and Jankovic....

A classification of inverse limit spaces of tent maps with periodic critical points

Lois Kailhofer (2003)

Fundamenta Mathematicae

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We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps $f_{a}$ , $f_{b}$ with periodic critical points, we show that the inverse limit spaces $(_{a}, f_{a})$ and $(_{b}, g_{b})$ are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.

Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment

Agnieszka Bogdewicz, Jerzy Grzybowski (2009)

Banach Center Publications

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Let $(ℝ, | | \cdot | |_{)}$ be a Minkowski space with a unit ball and let $ϱ_{H}^{}$ be the Hausdorff metric induced by ${| | \cdot | |}_{}$ in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to $ϱ_{H}^{B ⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(², ϱ_{H}^{})$ over any two-dimensional Minkowski space.

Some properties of perfect metric spaces

Angelo Bella, Biagio Ricceri (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questa Nota, dati uno spazio metrico perfetto $X$ ed un suo sottoinsieme $K$ chiuso e raro, si dimostra l'esistenza di una funzione continua $f:X\to[0,1]$ tale che $int(f^{-1}(t))=\emptyset$ per ogni $t\in[0,1]$ , $f(x)=0$ per ogni $x\in K$ e $f(y)=1$ per qualche $y\in X\setminus K$ . In particolare, ciò permette di dare risposta simultaneamente a due questioni poste in [2]. Si mettono in evidenza, poi, ulteriori conseguenze di tale risultato.

On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas (2011)

Fundamenta Mathematicae

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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{σ}$ . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0, σ}$ is continuous at σ₀ as the function of the parameter $σ \in \bar{₀}$ if and only if $H D (J_{0, σ ₀}) \geq 4 / 3$ . Since $H D (J_{0, σ}) > 4 / 3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $H D (J_{0, σ})$ on an open and dense subset of...

Szpilrajn type theorem for concentration dimension

Jozef Myjak, Tomasz Szarek (2002)

Fundamenta Mathematicae

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Let X be a locally compact, separable metric space. We prove that $d i m_{T} X = i n f d i m_{L} X^{'} : X^{'} i s h o m e o m o r p h i c t o X$ , where $d i m_{L} X$ and $d i m_{T} X$ stand for the concentration dimension and the topological dimension of X, respectively.

The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin, Sheila Sandon (2015)

Journal of the European Mathematical Society

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We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on $ℝ^{2 n} \times S^{1}$ and $ℝ P^{2 n + 1}$ . On the other hand we also show by elementary arguments...

The equation $\bar{\partial}u=f$ the intersection of pseudoconvex domains

Alessandro Perotti (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $C F L (_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $L I P (_{r})$ of equivalence classes of all real-valued Lipschitz maps on $_{r}$ . The space of all signed (real-valued) Borel measures on $_{r}$ is isometrically embedded in the dual space of $C F L (_{r})$ and it is shown that the image...

The Schreier Property and Gauss' Lemma

Daniel D. Anderson, Muhammad Zafrullah (2007)

Bollettino dell'Unione Matematica Italiana

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Let $D$ be an integral domain with quotient field $D$ . Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus\{0\}$ , $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$ . A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus\{0\}$ , there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_{v}$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$ . We also show that $D$ is Schreier if and only if every...

On the paradox of two $R_{1}, R_{2}$ domains in Schwarzschild exterior solution

A. Zięba (1972)

Colloquium Mathematicae

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The AR-Property of the spaces of closed convex sets

Katsuro Sakai, Masato Yaguchi (2006)

Colloquium Mathematicae

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Let $C o n v_{H} (X)$ , $C o n v_{A W} (X)$ and $C o n v_{W} (X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $C o n v_{H} (X)$ and the space $C o n v_{A W} (X)$ are AR. In case X is separable, $C o n v_{W} (X)$ is locally path-connected.

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $L C^{n - 1}$ -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki (2015)

Fundamenta Mathematicae

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Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W (ρ)}$ . It is known that $⟨ C L (X), τ_{W (ρ)} ⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨ C L (X), τ_{W (ρ)} ⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨ C L (X), τ_{W (ρ)} ⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces. ...

Remarks on $\mathcal{S}$ -Closedness in Topological Spaces

Zbigniew Duszyński (2007)

Bollettino dell'Unione Matematica Italiana

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Corresponding to [27], some properties of S-closed subspaces and subsets $\mathcal{S}$ -closed relative to a topological space are proved. Conditions under which mappings preserve certain $\mathcal{S}$ -closed subspaces are investigated.

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

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It is shown that the hyperspace $C l d_{H} (X)$ (resp. $B d d_{H} (X)$ ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

A two-dimensional univoque set

Martijn de Vrie, Vilmos Komornik (2011)

Fundamenta Mathematicae

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Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x = \sum_{i = 1}^{\infty} c_{i} q^{- i}$ with integer coefficients $c_{i}$ satisfying $0 \leq c_{i} < q$ , i ≥ 1. In this case we say that $(c_{i}) = c ₁ c ₂ . . .$ is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also...

Some properties of perfect metric spaces

Angelo Bella, Biagio Ricceri (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Isomorphisms of Royden Type Algebras Over $\mathbb{S}^{1}$

Teresa Radice, Eero Saksman, Gabriella Zecca (2009)

Bollettino dell'Unione Matematica Italiana

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Let $\mathbb{S}^{1}$ and $\mathbb{D}$ be the unit circle and the unit disc in the plane and let us denote by $\mathcal{A}(\mathbb{S}^{1})$ the algebra of the complex-valued continuous functions on $\mathbb{S}^{1}$ which are traces of functions in the Sobolev class $W^{1,2}(\mathbb{D})$ . On $\mathcal{A}(\mathbb{S}^{1})$ we define the following norm $\|u\|=\|u\|_{L^{\infty}(\mathbb{S}^{1})}+\left(\iint_{\mathbb{D}}|\nabla\tilde% {u}|^{2}\right)^{1/2}$ where is the harmonic extension of $u$ to $\mathbb{D}$ . We prove that every isomorphism of the functional algebra $\mathcal{A}(\mathbb{S}^{1})$ is a quasitsymmetric change of variables on $\mathbb{S}^{1}$ .