### A class of retracts in ${L}^{p}$ with some applications to differential inclusion

Grzegorz Bartuzel, Andrzej Fryszkowski (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Grzegorz Bartuzel, Andrzej Fryszkowski (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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A. Chigogidze (2013)

Fundamenta Mathematicae

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We characterize, in terms of X, the extensional dimension of the Stone-Čech corona βX∖X of a locally compact and Lindelöf space X. The non-Lindelöf case is also settled in terms of extending proper maps with values in ${I}^{\tau}\setminus L$, where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a ${Z}_{\tau}$-set X in the Tikhonov cube ${I}^{\tau}$ we find a necessary and sufficient condition, in terms of ${I}^{\tau}\setminus X$, for X to be in the class AE([L]). We also introduce a concept of a proper absolute...

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{k\to d}$ be the following algorithmic problem: Given a ﬁnite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${\mathbb{R}}^{d}$? Known results easily imply polynomiality of ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{d\to d}$ and ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{(d-1)\to d}$ are undecidable for each ${}_{d}\ge 5$. Our main result is NP-hardness of ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{2\to 4}$ and, more generally, of ${\mathrm{\U0001d674\U0001d67c\U0001d671\U0001d674\U0001d673}}_{k\to d}$ for all...

Franklin D. Tall (2002)

Fundamenta Mathematicae

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Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define ${X}_{M}$ to be X ∩ M with topology generated by $U\cap M:U\in \cap M$. Suppose ${X}_{M}$ is homeomorphic to the irrationals; must $X={X}_{M}$? We have partial results. We also answer a question of Gruenhage by showing that if ${X}_{M}$ is homeomorphic to the “Long Cantor Set”, then $X={X}_{M}$.

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into ${W}^{m,p}\left(\Omega \right)$ into ${L}^{S}\left(\Omega \right)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding ${W}^{m,p}\left(\Omega \right)$ into ${L}^{\phi}\left(\Omega \right)$...............................................................

Th. Friedrich (1974)

Colloquium Mathematicae

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Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from ${\ell}_{s}$ to ${\ell}_{p}\otimes \u0302{\ell}_{q}$ is compact when 1 ≤ p,q < s and that every operator from ${\ell}_{s}$ to ${\ell}_{p}\widehat{\otimes \u0302}{\ell}_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

D. W. Hajek, I. Irizarry (1981)

Matematički Vesnik

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J. Musiałek (1969)

Annales Polonici Mathematici

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Tadeusz Dobrowolski, Witold Marciszewski (2002)

Fundamenta Mathematicae

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In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an ${F}_{\sigma \delta \sigma}$-subset of X and contains a retract R so that $R\times {E}^{\omega}$ is not homeomorphic to ${E}^{\omega}$. This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

Hisao Kato, Eiichi Matsuhashi (2006)

Colloquium Mathematicae

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The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C(X,{I}^{p+2k+1-i})$ such that the diagonal product $f\times g:X\to Y\times {I}^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense ${G}_{\delta}$-subset of $C(X,{I}^{p+2k+1-i})$. In this paper, we prove that if f: X → Y is as above and ${D}_{j}$ (j = 1,..., k) are superdendrites, then the set of maps h in $C(X,{\prod}_{j=1}^{k}{D}_{j}\times {I}^{p+1-i})$ such that $f\times h:X\to Y\times \left({\prod}_{j=1}^{k}{D}_{j}\times {I}^{p+1-i}\right)$ is (i+1)-to-1 is a dense ${G}_{\delta}$-subset of $C(X,{\prod}_{j=1}^{k}{D}_{j}\times {I}^{p+1-i})$ for each 0 ≤ i ≤ p.

Tomasz Słonka (2012)

Colloquium Mathematicae

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We show that if n is a positive integer and ${2}^{\aleph \u2080}\le \aleph \u2099$, then for every positive integer m and for every real constant c > 0 there are functions $f\u2081,...,{f}_{n+m}:\mathbb{R}\u207f\to \mathbb{R}$ such that $(f\u2081,...,{f}_{n+m})\left(\mathbb{R}\u207f\right)={\mathbb{R}}^{n+m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $({f}_{i\u2081},...,{f}_{i\u2099})\left(y\right)=y+w$ for $y\in x+(-c,c)\times {\mathbb{R}}^{n-1}$.

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega}^{\omega}$ there exists a continuous function $f:{\omega}^{\omega}\to X$ such that ${f}^{-1}\left(C\u2080\right)=D\u2080$ and ${f}^{-1}\left(C\u2081\right)=D\u2081$. We give several explicit examples of complete pairs of coanalytic sets.

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the ${L}_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on ${L}_{p,q}\left(B\right)$........ 235. Examples in ${L}_{p,q}$ theory...................................................................................

Krzysztof Kołodziejczyk (1987)

Colloquium Mathematicae

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Hans Triebel (1994)

Studia Mathematica

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Let ${f}^{j}={\sum}_{k}{a}_{k}f({2}^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${\mathbb{R}}^{n}$ having integer-valued components, j∈ℕ and ${a}_{k}\in \u2102$. Let ${A}_{pq}^{s}$ be either ${B}_{pq}^{s}$ or ${F}_{pq}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${\mathbb{R}}^{n}.$ The aim of the paper is to clarify under what conditions $\parallel {f}^{j}|{A}_{pq}^{s}\parallel $ is equivalent to ${2}^{j(s-n/p)}({\sum}_{k}|{a}_{k}{|}^{p}{)}^{1/p}\parallel f|{A}_{pq}^{s}\parallel $.

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial ${x}^{n}-1$: “How often does ${x}^{n}-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when ${x}^{n}-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${\mathbb{F}}_{p}\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

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...

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let ${\mathcal{F}}_{h}^{i}(k,n)$ be the $i$-th ordered configuration space of all distinct points ${H}_{1},...,{H}_{h}$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${\scriptstyle {\u2102}^{n}}$, whose sum is a subspace of dimension $i$. We prove that ${\mathcal{F}}_{h}^{i}(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^{h}$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\>2$ and equal to the braid group of the sphere ${\scriptstyle \u2102}$ ${P}^{1}$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

Alessandro Caterino, Maria Cristina Vipera (1988)

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

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Let $F\subset {C}^{\ast}(X)$ be a vector sublattice over $\mathbb{R}$ which separates points from closed sets of $X$. The compactification ${e}_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice ${F}_{z}$ containing $F$ such that $\omega (Z(F))=\omega (Z({F}_{z}))={e}_{F}X$. In particular this implies that $\omega (Z(F))\ge {e}_{F}X$. Conditions in order to be $\omega (Z(F))={e}_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an...

Francesca Angrisani, Giacomo Ascione (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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Inspired by a result from Leibov, we ﬁnd that the supremum deﬁning the $BLO$ norm in $[0,1]$ is actually attained by a speciﬁc sub-interval of $[0,1]$ for $f\in VLO([0,1])$

Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik (2012)

Colloquium Mathematicae

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For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ${\sum}_{n=1}^{\infty}x\left(n\right)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of ${\sum}_{n=1}^{\infty}b\left(n\right)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....

R. Sinha (1973)

Matematički Vesnik

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Davinder Singh, Brij Kishore Tyagi, Jeetendra Aggarwal, Jogendra K. Kohli (2015)

Mathematica Bohemica

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A new class of functions called “${R}_{z}$-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of ${R}_{z}$-supercontinuous functions properly includes the class of ${R}_{\mathrm{cl}}$-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $\mathrm{cl}$-supercontinuous ($\equiv $ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983),...