The transfinite diameter of the real ball and simplex

T. Bloom; L. Bos; N. Levenberg

Displaying similar documents to “The transfinite diameter of the real ball and simplex”

Chebyshev series expansions of the functions $J_{v} (k x) / {(k x)}^{v}$ and $I_{v} (k x) / {(k x)}^{v}$

Z. Cylkowski (1966)

Applicationes Mathematicae

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On some properties of Chebyshev polynomials

Hacène Belbachir, Farid Bencherif (2008)

Discussiones Mathematicae - General Algebra and Applications

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Letting $T_{n}$ (resp. $U_{n}$ ) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ${(X^{k} T_{n - k})}_{k}$ and ${(X^{k} U_{n - k})}_{k}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space $_{n} [X]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_{n}$ and $U_{n}$ admit remarkableness integer coordinates on each of the two basis.

Discriminants of Chebyshev radical extensions

T. Alden Gassert (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $t$ be any integer and fix an odd prime $ℓ$ . Let $Φ (x) = T_{ℓ}^{n} (x) - t$ denote the $n$ -fold composition of the Chebyshev polynomial of degree $ℓ$ shifted by $t$ . If this polynomial is irreducible, let $K = ℚ (θ)$ , where $θ$ is a root of $Φ$ . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on $t$ that ensure $K$ is monogenic. For other values of $t$ , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of $K$ and compute an integral basis for the ring...

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne (2017)

Applications of Mathematics

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Acute triangles are defined by having all angles less than $π / 2$ , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension $n \geq 3$ , acuteness is defined by demanding that all dihedral angles between $(n - 1)$ -dimensional faces are smaller than $π / 2$ . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property...

Hardness of embedding simplicial complexes in $ℝ^{d}$

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let ${𝙴𝙼𝙱𝙴𝙳}_{k \to d}$ 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 $ℝ^{d}$ ? Known results easily imply polynomiality of ${𝙴𝙼𝙱𝙴𝙳}_{k \to 2}$ ( $k = 1, 2$ ; the case $k = 1, d = 2$ is graph planarity) and of ${𝙴𝙼𝙱𝙴𝙳}_{k \to 2 k}$ for all $k \geq 3$ . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${𝙴𝙼𝙱𝙴𝙳}_{d \to d}$ and ${𝙴𝙼𝙱𝙴𝙳}_{(d - 1) \to d}$ are undecidable for each $_{d} \geq 5$ . Our main result is NP-hardness of ${𝙴𝙼𝙱𝙴𝙳}_{2 \to 4}$ and, more generally, of ${𝙴𝙼𝙱𝙴𝙳}_{k \to d}$ for all...

Explicit extension maps in intersections of non-quasi-analytic classes

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

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We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces $_{()} ({[- 1, 1]}^{r})$ ; (b) there is no continuous linear extension map from $Λ_{()}^{(r)}$ into $_{()} (ℝ^{r})$ ; (c) under some additional assumption on , there is an explicit extension map from $_{()} ({[- 1, 1]}^{r})$ into $_{()} ({[- 2, 2]}^{r})$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

A characterization of sets in $ℝ^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček (2022)

Czechoslovak Mathematical Journal

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We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

Counting triangles that share their vertices with the unit $n$ -cube

Brandts, Jan, Cihangir, Apo

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This paper is about $0 / 1$ -triangles, which are the simplest nontrivial examples of $0 / 1$ -polytopes: convex hulls of a subset of vertices of the unit $n$ -cube $I^{n}$ . We consider the subclasses of right $0 / 1$ -triangles, and acute $0 / 1$ -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of $I^{n}$ .

A characterization of the disc $D^{n}$

Th. Friedrich (1974)

Colloquium Mathematicae

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Complexity of the method of averaging

Dalík, Josef

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The general method of averaging for the superapproximation of an arbitrary partial derivative of a smooth function in a vertex $a$ of a simplicial triangulation $𝒯$ of a bounded polytopic domain in $ℜ^{d}$ for any $d \geq 2$ is described and its complexity is analysed.

The norm of the polynomial truncation operator on the unit disk and on [-1,1]

Tamás Erdélyi (2001)

Colloquium Mathematicae

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Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. $ₙ^{c}$ ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $P ₙ \in ₙ^{c}$ of the form $P ₙ (z) : = \sum_{j = 0}^{n} a_{j} z_{j}$ , $a_{j} \in C$ , by $S ₙ (P ₙ) (z) : = \sum_{j = 0}^{n} a ̃_{j} z_{j}$ , $a ̃_{j} : = a_{j} | a_{j} | m i n | a_{j} |, 1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $∥ S ₙ ∥_{\infty, \partial D}^{r e a l} : = s u p_{P ₙ \in ₙ} (m a x_{z \in \partial D} | S ₙ (P ₙ) (z) |) / (m a x_{z \in \partial D} | P ₙ (z) |)$ , $∥ S ₙ ∥_{\infty, \partial D}^{c o m p} : = s u p_{P ₙ \in ₙ^{c}} (m a x_{z \in \partial D} | S ₙ (P ₙ) (z) |) / (m a x_{z \in \partial D} | P ₙ (z) |$ . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

On almost everywhere differentiability of the metric projection on closed sets in $l^{p} (ℝ^{n})$ , $2 < p < \infty$

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

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Let $F$ be a closed subset of $ℝ^{n}$ and let $P (x)$ denote the metric projection (closest point mapping) of $x \in ℝ^{n}$ onto $F$ in $l^{p}$ -norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $ℝ^{n}$ in the Euclidean case $p = 2$ . We consider the case $2 < p < \infty$ and prove that the $i$ th component $P_{i} (x)$ of $P (x)$ is differentiable a.e. if $P_{i} (x) \neq x_{i}$ and satisfies Hölder condition of order $1 / (p - 1)$ if $P_{i} (x) = x_{i}$ .

Mobius invariant Besov spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We give new characterizations of the analytic Besov spaces $B_{p}$ on the unit ball $𝔹$ of $ℂ^{n}$ in terms of oscillations and integral means over some Euclidian balls contained in $𝔹$ .

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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

Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

Albert Baernstein II, Robert C. Culverhouse (2002)

Studia Mathematica

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Let $X = \sum_{i = 1}^{k} a_{i} U_{i}$ , $Y = \sum_{i = 1}^{k} b_{i} U_{i}$ , where the $U_{i}$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_{i}, b_{i}$ are real constants. We prove that if $b ²_{i}$ is majorized by $a ²_{i}$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L ² - L^{p}$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

On starshapedness of the union of closed sets in $R^{n}$

Krzysztof Kołodziejczyk (1987)

Colloquium Mathematicae

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