Chebyshev Distance

Roland Coghetto

Displaying similar documents to “Chebyshev Distance”

Sign changes in $π_{q, a} (x) - π_{q, b} (x)$

Kevin Ford, Richard H. Hudson (2001)

Acta Arithmetica

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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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Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond, Wen-Bin Zhang (2013)

Acta Arithmetica

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If the counting function N(x) of integers of a Beurling generalized number system satisfies both $\int_{1}^{\infty} x^{- 2} | N (x) - A x | d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (1)$ , then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that $\int_{1}^{\infty} | N (x) - A x | x^{- 2} d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (f (x))$ do not imply the Chebyshev bound.

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.

The transfinite diameter of the real ball and simplex

T. Bloom, L. Bos, N. Levenberg (2012)

Annales Polonici Mathematici

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We calculate the transfinite diameter for the real unit ball $B_{d} : = x \in ℝ^{d} : | x | \leq 1$ and the real unit simplex $T_{d} : = x \in ℝ_{+}^{d} : \sum_{j = 1}^{d} x_{j} \leq 1 .$

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].

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 $x \in C$ .

Kolmogorov problem in $W^{r} H^{ω} [0, 1]$ and extremal Zolotarev ω-splines

Bagdasarov Sergey K.

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AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) $f^{(m)} (ξ) \to s u p$ , $f \in W^{r} H^{ω} [ξ, b]$ , ${| | f | |}_{ℂ [a, b]} \leq B$ ,for all 0 < m ≤ r, ξ ≤ a or ξ = (a+b)/2, all B > 0 and concave moduli of continuity ω on ℝ₊. It is shown that any extremal function $=_{B, r, m, ω, ξ}$ of the problem (*) enjoys the following two characteristic properties. First, the function $^{(r)} (\cdot) -^{(r)} (ξ)$ is extremal for the problem(**) $\int_{ξ}^{b} h (t) ψ (t) d t \to s u p$ , $h \in H^{ω} [ξ, b]$ , h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

Algebraic and topological properties of some sets in ℓ₁

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 $\sum_{n = 1}^{\infty} x (n)$ . 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 $\sum_{n = 1}^{\infty} b (n)$ 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....

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

On biorthogonal systems whose functionals are finitely supported

Christina Brech, Piotr Koszmider (2011)

Fundamenta Mathematicae

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We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2 n}$ such that $C (K_{2 n})$ has no uncountable (semi)biorthogonal sequence ${(f_{ξ}, μ_{ξ})}_{ξ \in ω ₁}$ where $μ_{ξ}$ ’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2 n}$ , but has biorthogonal systems ${(f_{ξ}, μ_{ξ})}_{ξ \in ω ₁}$ where $μ_{ξ}$ ’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves...

A new Lindelöf space with points $G_{δ}$

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that $♦^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{ℵ_{1}}$ which has points $G_{δ}$ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

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Let $Q = {(q_{n})}_{n = 1}^{\infty}$ be a sequence of bases with $q_{i} \geq 2$ . In the case when the $q_{i}$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$ -Cantor series expansion is both $Q$ -normal and $Q$ -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$ , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.

Complete pairs of coanalytic sets

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 $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a zero-dimensional space and $C_{c} (X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_{c}^{K} (X)$ (resp., $C_{c}^{ψ} (X))$ the set of all functions in $C_{c} (X)$ with compact (resp., pseudocompact) support. First, we observe that $C_{c}^{K} (X) = O_{c}^{β_{0} X ∖ X}$ (resp., $C_{c}^{ψ} (X) = M_{c}^{β_{0} X ∖ υ_{0} X}$ ), where $β_{0} X$ is the Banaschewski compactification of $X$ and $υ_{0} X$ is the $ℕ$ -compactification of $X$ . This implies that for an $ℕ$ -compact space $X$ , the intersection of all free maximal ideals in $C_{c} (X)$ is equal to $C_{c}^{K} (X)$ , i.e., $M_{c}^{β_{0} X ∖ X} = C_{c}^{K} (X)$ . By applying...