## Displaying similar documents to “Optimality of Chebyshev bounds for Beurling generalized numbers”

Acta Arithmetica

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### Chebyshev Distance

Formalized Mathematics

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In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of [...] ℰTn ${ℰ}_{T}^{n}$ and in [20] he has formalized that [...] ℰTn ${ℰ}_{T}^{n}$ is second-countable, we build (in the topological sense defined in [23]) a denumerable base of [...] ℰTn ${ℰ}_{T}^{n}$ . Then we introduce the n-dimensional intervals (interval in n-dimensional Euclidean space, pavé (borné) de ℝn [16], semi-intervalle (borné) de ℝn [22]). We conclude with the definition of Chebyshev distance [11]. ...

### Chebyshev series expansions of the functions ${J}_{v}\left(kx\right)/{\left(kx\right)}^{v}$ and ${I}_{v}\left(kx\right)/{\left(kx\right)}^{v}$

Applicationes Mathematicae

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### On some properties of Chebyshev polynomials

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 ${\left({X}^{k}{T}_{n-k}\right)}_{k}$ and ${\left({X}^{k}{U}_{n-k}\right)}_{k}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space ${}_{n}\left[X\right]$ 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

Annales Polonici Mathematici

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We calculate the transfinite diameter for the real unit ball ${B}_{d}:=x\in {ℝ}^{d}:|x|\le 1$ and the real unit simplex ${T}_{d}:=x\in {ℝ}_{+}^{d}:{\sum }_{j=1}^{d}{x}_{j}\le 1.$

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

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 ${}_{\left(\right)}\left({\left[-1,1\right]}^{r}\right)$; (b) there is no continuous linear extension map from ${\Lambda }_{\left(\right)}^{\left(r\right)}$ into ${}_{\left(\right)}\left({ℝ}^{r}\right)$; (c) under some additional assumption on , there is an explicit extension map from ${}_{\left(\right)}\left({\left[-1,1\right]}^{r}\right)$ into ${}_{\left(\right)}\left({\left[-2,2\right]}^{r}\right)$ 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

Journal de Théorie des Nombres de Bordeaux

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Let $t$ be any integer and fix an odd prime $\ell$. Let $\Phi \left(x\right)={T}_{\ell }^{n}\left(x\right)-t$ denote the $n$-fold composition of the Chebyshev polynomial of degree $\ell$ shifted by $t$. If this polynomial is irreducible, let $K=ℚ\left(\theta \right)$, where $\theta$ is a root of $\Phi$. 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

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 $\left(X,\parallel ·\parallel \right)$ containing asymptotically isometric copy of the space ${c}_{0}$ there is a bounded, closed and convex set $C\subset X$ with the Chebyshev radius $r\left(C\right)=1$ such that for every $k\ge 1$ there exists a $k$-contractive mapping $T:C\to C$ with $\parallel x-Tx\parallel >1-1/k$ for any $x\in C$.

### On sum-product representations in ${ℤ}_{q}$

Journal of the European Mathematical Society

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The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${ℤ}_{q}$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $|A|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the...

### Nilakantha's accelerated series for π

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 $\pi ={\sum }_{n=0}^{\infty }\left(\left(5n+3\right)n!\left(2n\right)!\right)/\left({2}^{n-1}\left(3n+2\right)!\right)$ with convergence as $13.{5}^{-n}$, in much the same way as the Euler transformation gives $\pi ={\sum }_{n=0}^{\infty }\left({2}^{n+1}n!n!\right)/\left(2n+1\right)!$ 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...

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

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AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) ${f}^{\left(m\right)}\left(\xi \right)\to sup$, $f\in {W}^{r}{H}^{\omega }\left[\xi ,b\right]$, ${||f||}_{ℂ\left[a,b\right]}\le B$,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 ${=}_{B,r,m,\omega ,\xi }$ of the problem (*) enjoys the following two characteristic properties. First, the function ${}^{\left(r\right)}\left(·\right){-}^{\left(r\right)}\left(\xi \right)$ is extremal for the problem(**) ${\int }_{\xi }^{b}h\left(t\right)\psi \left(t\right)dt\to sup$, $h\in {H}^{\omega }\left[\xi ,b\right]$, h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...

### A non-Chebyshev finite-dimensional subspace in ${H}_{1}$

Banach Center Publications

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### Lower bounds for the largest eigenvalue of the gcd matrix on $\left\{1,2,\cdots ,n\right\}$

Czechoslovak Mathematical Journal

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Consider the $n×n$ matrix with $\left(i,j\right)$’th entry $gcd\left(i,j\right)$. Its largest eigenvalue ${\lambda }_{n}$ and sum of entries ${s}_{n}$ satisfy ${\lambda }_{n}>{s}_{n}/n$. Because ${s}_{n}$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that ${\lambda }_{n}>6{\pi }^{-2}nlogn$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969).

### Some characterizations of the class ${}_{m}\left(\Omega \right)$ and applications

Annales Polonici Mathematici

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We give some characterizations of the class ${}_{m}\left(\Omega \right)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

### Upper bounds for singular perturbation problems involving gradient fields

Journal of the European Mathematical Society

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We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy ${E}_{\epsilon }\left(v\right)=\epsilon {\int }_{\Omega }|{\nabla }^{2}{v|}^{2}dx+{\epsilon }^{-1}{\int }_{\Omega }{\left(1-|\nabla v|}^{2}{\right)}^{2}dx$ over $v\in {H}^{2}\left(\Omega \right)$, where $\epsilon >0$ is a small parameter. Given $v\in {W}^{1,\infty }\left(\Omega \right)$ such that $\nabla v\in BV$ and $|\nabla v|=1$ a.e., we construct a family $\left\{{v}_{\epsilon }\right\}$ satisfying: ${v}_{\epsilon }\to v$ in ${W}^{1,p}\left(\Omega \right)$ and ${E}_{\epsilon }\left({v}_{\epsilon }\right)\to \frac{1}{3}{\int }_{{J}_{\nabla v}}{|{\nabla }^{+}v-{\nabla }^{-}v|}^{3}d{ℋ}^{N-1}$ as $\epsilon$ goes to 0.

### On a system of equations with primes

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n\ge 3$, let ${u}_{1},...,{u}_{n}$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\left\{1,...,n\right\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \left\{±1\right\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_{i}-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide ${u}_{1}\cdots {u}_{n}$? We answer this question in the positive when the ${u}_{i}$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_{0}\in \left\{±1\right\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

### A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Acta Arithmetica

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Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/|G|}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. We prove, among other results, that $₁\left({C}_{n₁}\oplus {C}_{n₂}\right)=n₁+n₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

### A note on representation functions with different weights

Colloquium Mathematicae

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For any positive integer k and any set A of nonnegative integers, let ${r}_{1,k}\left(A,n\right)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ and ${r}_{1,l}\left(A,n\right)={r}_{1,l}\left(ℕ\setminus A,n\right)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ for all n ≥ n₀, we have ${r}_{1,k}\left(A,n\right)\to \infty$ as n → ∞.

### Dimension of weakly expanding points for quadratic maps

Bulletin de la Société Mathématique de France

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For the real quadratic map ${P}_{a}\left(x\right)={x}^{2}+a$ and a given $ϵ>0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with ${P}_{a}^{n}{|}_{J}$ univalent, with bounded distortion and $B\left(0,ϵ\right)\subseteq {P}_{a}^{n}\left(J\right)$ for some $n\in ℕ$. The $ϵ$-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha$ denote the negative fixed point and $M$ the first return time of the critical orbit to $\left[\alpha ,-\alpha \right]$. We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

### Towards Bauer's theorem for linear recurrence sequences

Colloquium Mathematicae

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Consider a recurrence sequence ${\left({x}_{k}\right)}_{k\in ℤ}$ of integers satisfying ${x}_{k+n}={a}_{n-1}{x}_{k+n-1}+...+a₁{x}_{k+1}+a₀{x}_{k}$, where $a₀,a₁,...,{a}_{n-1}\in ℤ$ are fixed and a₀ ∈ -1,1. Assume that ${x}_{k}>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that ${x}_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|{x}_{k}$ for some k ∈ ℕ and (-D/q) = -1.

### Sums of positive density subsets of the primes

Acta Arithmetica

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We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $\left(1-o\left(1\right)\right)\alpha /\left({e}^{\gamma }loglog\left(1/\beta \right)\right)$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ{*}_{m}$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A,B\subseteq ℤ{*}_{m}$ of densities α...

### A quantitative aspect of non-unique factorizations: the Narkiewicz constants II

Colloquium Mathematicae

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Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves, for x → ∞, asymptotically like $x{\left(logx\right)}^{1/|G|-1}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. In this article, it is proved that for every prime p, $₁\left({C}_{p}\oplus {C}_{p}\right)=2p$, and it is also proved that $₁\left({C}_{mp}\oplus {C}_{mp}\right)=2mp$ if $₁\left({C}_{m}\oplus {C}_{m}\right)=2m$ and m is large enough. In particular, it is shown...

### Property C for ODE and Applications to an Inverse Problem for a Heat Equation

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ${\ell }_{j}:=-d²/dx²+k²{q}_{j}\left(x\right)$, k = const > 0, j = 1,2, $0. Suppose that (*) ${\int }_{0}^{1}p\left(x\right)u₁\left(x,k\right)u₂\left(x,k\right)dx=0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and ${u}_{j}$ solves the problem ${\ell }_{j}{u}_{j}=0$, 0 ≤ x ≤ 1, ${u}_{j}^{\text{'}}\left(0,k\right)=0$, ${u}_{j}\left(0,k\right)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

### Limits of log canonical thresholds

Annales scientifiques de l'École Normale Supérieure

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Let ${𝒯}_{n}$ denote the set of log canonical thresholds of pairs $\left(X,Y\right)$, with $X$ a nonsingular variety of dimension $n$, and $Y$ a nonempty closed subscheme of $X$. Using non-standard methods, we show that every limit of a decreasing sequence in ${𝒯}_{n}$ lies in ${𝒯}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${𝒯}_{n}$ is closed in $𝐑$; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

### Uniform algebras and analytic multi­functions

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

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Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_{A}/f(\partial_{A})$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda\to\log\max g(f^{-1}(\lambda))$ è subarmonica su $G$ e che l’applicazione $\lambda\to g(f^{-1}(\lambda))$ è analitica nel senso di Oka.

### Moser's Inequality for a class of integral operators

Studia Mathematica

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Let 1 < p < ∞, q = p/(p-1) and for $f\in {L}^{p}\left(0,\infty \right)$ define $F\left(x\right)=\left(1/x\right){ʃ}_{0}^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant ${C}_{p}$ such that $su{p}_{a\le 1}su{p}_{f\in {B}_{p}}{ʃ}_{0}^{\infty }exp\left[a{x}^{q}{|F\left(x\right)|}^{q}-x\right]dx={C}_{p}$ where ${B}_{p}$ is the unit ball of ${L}^{p}$. Moreover, the value a = 1 is sharp. We observe that $F={K}_{1}$ f where the integral operator ${K}_{1}$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...