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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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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.
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]. ...
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 and , 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 and do not imply the Chebyshev bound.
Z. Cylkowski (1966)
Applicationes Mathematicae
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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. 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
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...
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...
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 .
Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)
Acta Arithmetica
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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in of degree d for which s consecutive coefficients are fixed. Our estimate asserts that , where . We also prove that , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of of degree d with s consecutive coefficients fixed as above. Finally, we show that , where ₂(d,0) denotes the average second moment for...
Joe Callaghan (2007)
Annales Polonici Mathematici
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Let K be any subset of . We define a pluricomplex Green’s function for θ-incomplete polynomials. We establish properties of analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute when K is a compact...
Wojciech Banaszczyk, Artur Lipnicki (2015)
Annales Polonici Mathematici
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The paper deals with the approximation by polynomials with integer coefficients in , 1 ≤ p ≤ ∞. Let be the space of polynomials of degree ≤ n which are divisible by the polynomial , r ≥ 0, and let be the set of polynomials with integer coefficients. Let be the maximal distance of elements of from in . We give rather precise quantitative estimates of for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of for p ≠ 2. It follows that as n → ∞. The results...
Katarzyna Grasela (2010)
Banach Center Publications
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We consider the space of ultradifferentiable functions with compact supports and the space of polynomials on . A description of the space of polynomial ultradistributions as a locally convex direct sum is given.
Umberto Bartocci, Maria Cristina Vipera (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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If is a polynomial with coefficients in the field of complex numbers, of positive degree , then has at least one root a with the following property: if , where is the multiplicity of , then (such a root is said to be a "free" root of ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree ) with coefficients in a field of positive characteristic (Sudbery's Conjecture). In this paper it...
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...
Peter Borwein, Tamás Erdélyi, Géza Kós (2013)
Acta Arithmetica
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For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≠ 0 of the form , 1/paj ∈ ℂsuch that divides P(x). For n ∈ ℕ and L > 0 let be the largest possible value of k for which there is a polynomial P ≠ 0 of the form , , , such that divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...
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. ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials of the form , , by , (here 0/0 is interpreted as 1). We define the norms of the truncation operators by , . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...
(2016)
Acta Arithmetica
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For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≢ 0 of the form , , , such that divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that . We find the size of and for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...
Umberto Bartocci, Maria Cristina Vipera (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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If is a polynomial with coefficients in the field of complex numbers, of positive degree , then has at least one root a with the following property: if , where is the multiplicity of , then (such a root is said to be a "free" root of ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree ) with coefficients in a field of positive characteristic (Sudbery's Conjecture). In this paper it...
Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be the uniform norm in the unit disk. We study the quantities where the infimum is taken over all polynomials of degree with and . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that . We find the exact values of and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
Fabiano G. B. Brito, Pablo M. Chacón, David L. Johnson (2008)
Bulletin de la Société Mathématique de France
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We establish in this paper a lower bound for the volume of a unit vector field defined on , . This lower bound is related to the sum of the absolute values of the indices of at and .
Joshua Harrington, Andrew Vincent, Daniel White (2013)
Journal de Théorie des Nombres de Bordeaux
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In this paper we investigate the factorization of the polynomials in the special case where is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that is monic and linear.
Yousef Zamani, Mahin Ranjbari (2018)
Czechoslovak Mathematical Journal
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Let be the complex vector space of homogeneous linear polynomials in the variables . Suppose is a subgroup of , and is an irreducible character of . Let be the symmetry class of polynomials of degree with respect to and . For any linear operator acting on , there is a (unique) induced operator acting on symmetrized decomposable polynomials by In this paper, we show that the representation of the general linear group is equivalent to the direct sum of copies...
Christophe Debry (2014)
Journal de Théorie des Nombres de Bordeaux
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Let be a cubic, monic and separable polynomial over a field of characteristic and let be the elliptic curve given by . In this paper we prove that the coefficient at in the –th division polynomial of equals the coefficient at in . For elliptic curves over a finite field of characteristic , the first coefficient is zero if and only if is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...