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Kevin Ford, Richard H. Hudson (2001)
Acta Arithmetica
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Kevin Ford, Richard H. Hudson (2001)
Acta Arithmetica
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Roland Coghetto (2016)
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 and in [20] he has formalized that [...] ℰTn is second-countable, we build (in the topological sense defined in [23]) a denumerable base of [...] ℰTn . 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]. ...
Boaz Cohen (2021)
Czechoslovak Mathematical Journal
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We shall describe how to construct a fundamental solution for the Pell equation over finite fields of characteristic . Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation .
Z. Cylkowski (1966)
Applicationes Mathematicae
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Hacène Belbachir, Farid Bencherif (2008)
Discussiones Mathematicae - General Algebra and Applications
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Letting (resp. ) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences and for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also and admit remarkableness integer coordinates on each of the two basis.
T. Bloom, L. Bos, N. Levenberg (2012)
Annales Polonici Mathematici
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We calculate the transfinite diameter for the real unit ball and the real unit simplex
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 ; (b) there is no continuous linear extension map from into ; (c) under some additional assumption on , there is an explicit extension map from into 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
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Let be any integer and fix an odd prime . Let denote the -fold composition of the Chebyshev polynomial of degree shifted by . If this polynomial is irreducible, let , where is a root of . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on that ensure is monogenic. For other values of , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of and compute an integral basis for the ring...
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 containing asymptotically isometric copy of the space there is a bounded, closed and convex set with the Chebyshev radius such that for every there exists a -contractive mapping with for any .
Mei-Chu Chang (2006)
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 , by performing sum and product set operations starting from a given subset of . We consider the case of very small sets and composite for which not much seemed known (nontrivial results were recently obtained when is prime or when log ). Roughly speaking we show that all residue classes are obtained from a -fold sum of an -fold product set of , where and , provided the...
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 with convergence as , in much the same way as the Euler transformation gives with convergence as . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...
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(*) , , ,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 of the problem (*) enjoys the following two characteristic properties. First, the function is extremal for the problem(**) , , h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...
S. B. Stechkin (1989)
Banach Center Publications
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Jorma K. Merikoski (2016)
Czechoslovak Mathematical Journal
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Consider the matrix with ’th entry . Its largest eigenvalue and sum of entries satisfy . Because cannot be expressed algebraically as a function of , 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 for all . If is large enough, this follows from F. Balatoni (1969).
Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)
Annales Polonici Mathematici
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We give some characterizations of the class 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
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We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy over , where is a small parameter. Given such that and a.e., we construct a family satisfying: in and as goes to 0.
Paolo Leonetti, Salvatore Tringali (2014)
Journal de Théorie des Nombres de Bordeaux
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Given an integer , let be pairwise coprime integers , a family of nonempty proper subsets of with “enough” elements, and a function . Does there exist at least one prime such that divides for some , but it does not divide ? We answer this question in the positive when the are prime powers and and are subjected to certain restrictions. We use the result to prove that, if and is a set of three or more primes that contains all prime divisors of any...
Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)
Acta Arithmetica
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Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves for x → ∞ asymptotically like . We prove, among other results, that for all integers n₁,n₂ with 1 < n₁|n₂.
Zhenhua Qu (2016)
Colloquium Mathematicae
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For any positive integer k and any set A of nonnegative integers, let 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 and 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 for all n ≥ n₀, we have as n → ∞.
Samuel Senti (2003)
Bulletin de la Société Mathématique de France
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For the real quadratic map and a given a point has good expansion properties if any interval containing also contains a neighborhood of with univalent, with bounded distortion and for some . The -weakly expanding set is the set of points which do not have good expansion properties. Let denote the negative fixed point and the first return time of the critical orbit to . We show there is a set of parameters with positive Lebesgue measure for which the Hausdorff...
Mariusz Skałba (2003)
Colloquium Mathematicae
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Consider a recurrence sequence of integers satisfying , where are fixed and a₀ ∈ -1,1. Assume that for all sufficiently large k. If there exists k₀∈ ℤ such that then for each negative integer -D there exist infinitely many rational primes q such that for some k ∈ ℕ and (-D/q) = -1.
Kaisa Matomäki (2013)
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 , 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 using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any of densities α...
Weidong Gao, Yuanlin Li, Jiangtao Peng (2011)
Colloquium Mathematicae
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Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves, for x → ∞, asymptotically like . In this article, it is proved that for every prime p, , and it is also proved that if and m is large enough. In particular, it is shown...