### Sign changes in ${\pi}_{q,a}\left(x\right)-{\pi}_{q,b}\left(x\right)$

Kevin Ford, Richard H. Hudson (2001)

Acta Arithmetica

Similarity:

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Kevin Ford, Richard H. Hudson (2001)

Acta Arithmetica

Similarity:

Roland Coghetto (2016)

Formalized Mathematics

Similarity:

In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of [...] ℰTn ${\mathcal{E}}_{T}^{n}$ and in [20] he has formalized that [...] ℰTn ${\mathcal{E}}_{T}^{n}$ is second-countable, we build (in the topological sense defined in [23]) a denumerable base of [...] ℰTn ${\mathcal{E}}_{T}^{n}$ . 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]. ...

Z. Cylkowski (1966)

Applicationes Mathematicae

Similarity:

Hacène Belbachir, Farid Bencherif (2008)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

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.

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

Annales Polonici Mathematici

Similarity:

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

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

Similarity:

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 ${}_{\left(\right)}\left({[-1,1]}^{r}\right)$; (b) there is no continuous linear extension map from ${\Lambda}_{\left(\right)}^{\left(r\right)}$ into ${}_{\left(\right)}\left({\mathbb{R}}^{r}\right)$; (c) under some additional assumption on , there is an explicit extension map from ${}_{\left(\right)}\left({[-1,1]}^{r}\right)$ into ${}_{\left(\right)}\left({[-2,2]}^{r}\right)$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

T. Alden Gassert (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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 $K=\mathbb{Q}\left(\theta \right)$, 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...

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

The aim of this paper is to show that for every Banach space $(X,\parallel \xb7\parallel )$ 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 $x\in C$.

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

Similarity:

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 ${\mathbb{Z}}_{q}$. 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 $\left|A\right|\sim logq$). 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...

David Brink (2015)

Acta Arithmetica

Similarity:

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

Bagdasarov Sergey K.

Similarity:

AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) ${f}^{\left(m\right)}\left(\xi \right)\to sup$, $f\in {W}^{r}{H}^{\omega}[\xi ,b]$, ${\left|\right|f\left|\right|}_{\u2102[a,b]}\le B$,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 ${}^{\left(r\right)}\left(\xb7\right){-}^{\left(r\right)}\left(\xi \right)$ is extremal for the problem(**) ${\int}_{\xi}^{b}h\left(t\right)\psi \left(t\right)dt\to sup$, $h\in {H}^{\omega}[\xi ,b]$, h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...

S. B. Stechkin (1989)

Banach Center Publications

Similarity:

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

Similarity:

Consider the $n\times n$ matrix with $(i,j)$’th entry $gcd(i,j)$. 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).

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

Similarity:

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.

Arkady Poliakovsky (2007)

Journal of the European Mathematical Society

Similarity:

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

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Given an integer $n\ge 3$, let ${u}_{1},...,{u}_{n}$ be pairwise coprime integers $\ge 2$, $\mathcal{D}$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon $ a function $\mathcal{D}\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod}_{i\in I}{u}_{i}-\epsilon \left(I\right)$ for some $I\in \mathcal{D}$, 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 $\mathcal{D}$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon}_{0}\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)

Acta Arithmetica

Similarity:

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 $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. We prove, among other results, that $\u2081\left({C}_{n\u2081}\oplus {C}_{n\u2082}\right)=n\u2081+n\u2082$ for all integers n₁,n₂ with 1 < n₁|n₂.

Zhenhua Qu (2016)

Colloquium Mathematicae

Similarity:

For any positive integer k and any set A of nonnegative integers, let ${r}_{1,k}(A,n)$ 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 ${r}_{1,k}(A,n)={r}_{1,k}(\mathbb{N}\setminus A,n)$ and ${r}_{1,l}(A,n)={r}_{1,l}(\mathbb{N}\setminus A,n)$ 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 ${r}_{1,k}(A,n)={r}_{1,k}(\mathbb{N}\setminus A,n)$ for all n ≥ n₀, we have ${r}_{1,k}(A,n)\to \infty $ as n → ∞.

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

Similarity:

For the real quadratic map ${P}_{a}\left(x\right)={x}^{2}+a$ and a given $\u03f5\>0$ 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 $B(0,\u03f5)\subseteq {P}_{a}^{n}\left(J\right)$ for some $n\in \mathbb{N}$. The $\u03f5$-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 $[\alpha ,-\alpha ]$. We show there is a set $\mathcal{R}$ of parameters with positive Lebesgue measure for which the Hausdorff...

Mariusz Skałba (2003)

Colloquium Mathematicae

Similarity:

Consider a recurrence sequence ${\left({x}_{k}\right)}_{k\in \mathbb{Z}}$ of integers satisfying ${x}_{k+n}={a}_{n-1}{x}_{k+n-1}+...+a\u2081{x}_{k+1}+a\u2080{x}_{k}$, where $a\u2080,a\u2081,...,{a}_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that ${x}_{k}>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that ${x}_{k\u2080}<0$ 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.

Kaisa Matomäki (2013)

Acta Arithmetica

Similarity:

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 $(1-o\left(1\right))\alpha /\left({e}^{\gamma}loglog(1/\beta )\right)$, 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 $\mathbb{Z}{*}_{m}$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A,B\subseteq \mathbb{Z}{*}_{m}$ of densities α...

Weidong Gao, Yuanlin Li, Jiangtao Peng (2011)

Colloquium Mathematicae

Similarity:

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 $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. In this article, it is proved that for every prime p, $\u2081\left({C}_{p}\oplus {C}_{p}\right)=2p$, and it is also proved that $\u2081\left({C}_{mp}\oplus {C}_{mp}\right)=2mp$ if $\u2081\left({C}_{m}\oplus {C}_{m}\right)=2m$ and m is large enough. In particular, it is shown...

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let ${\ell}_{j}:=-d\xb2/dx\xb2+k\xb2{q}_{j}\left(x\right)$, k = const > 0, j = 1,2, $0<essinf{q}_{j}\left(x\right)\le esssup{q}_{j}\left(x\right)<\infty $. Suppose that (*) ${\int}_{0}^{1}p\left(x\right)u\u2081(x,k)u\u2082(x,k)dx=0$ 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, ${u}_{j}^{\text{'}}(0,k)=0$, ${u}_{j}(0,k)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Tommaso de Fernex, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let ${\mathcal{T}}_{n}$ denote the set of log canonical thresholds of pairs $(X,Y)$, 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 ${\mathcal{T}}_{n}$ lies in ${\mathcal{T}}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${\mathcal{T}}_{n}$ is closed in $\mathbf{R}$; 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...

Zbigniew Slodkowski (1983)

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

Similarity:

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 $\lambda \to \mathrm{log}\mathrm{max}g({f}^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g({f}^{-1}(\lambda ))$ è analitica nel senso di Oka.

Finbarr Holland, David Walsh (1995)

Studia Mathematica

Similarity:

Let 1 < p < ∞, q = p/(p-1) and for $f\in {L}^{p}(0,\infty )$ define $F\left(x\right)=(1/x){\u0283}_{0}^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant ${C}_{p}$ such that $su{p}_{a\le 1}su{p}_{f\in {B}_{p}}{\u0283}_{0}^{\infty}exp[a{x}^{q}{\left|F\left(x\right)\right|}^{q}-x]dx={C}_{p}$ 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...