Displaying similar documents to “A generalized Kahane-Khinchin inequality”

On sum-product representations in q

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 q , by performing sum and product set operations starting from a given subset A of 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 | log q ). 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 log q and log k log q , provided the...

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

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For the real quadratic map P a ( x ) = 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 ( 0 , ϵ ) P a n ( J ) for some n . The ϵ -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [ α , - α ] . We show there is a set of parameters with positive Lebesgue measure for which the Hausdorff...

On the divisor function over Piatetski-Shapiro sequences

Hui Wang, Yu Zhang (2023)

Czechoslovak Mathematical Journal

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Let [ x ] be an integer part of x and d ( n ) be the number of positive divisor of n . Inspired by some results of M. Jutila (1987), we prove that for 1 < c < 6 5 , n x d ( [ n c ] ) = c x log x + ( 2 γ - c ) x + O x log x , where γ is the Euler constant and [ n c ] is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

Limits of log canonical thresholds

Tommaso de Fernex, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

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Let 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 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...

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

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 K be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let F k ( x ) 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 ( x ) behaves for x → ∞ asymptotically like x ( l o g x ) 1 - 1 / | G | ( l o g l o g x ) k ( G ) . We prove, among other results, that ( C n C n ) = n + n for all integers n₁,n₂ with 1 < n₁|n₂.

Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

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We prove the L p boundedness of the Marcinkiewicz integral operators μ Ω on n × × n k under the condition that Ω L ( l o g L ) k / 2 ( n - 1 × × n k - 1 ) . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

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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 ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( 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 ( A , n ) for all n ≥ n₀, we have r 1 , k ( A , n ) as n → ∞.

Inequalities for Taylor series involving the divisor function

Horst Alzer, Man Kam Kwong (2022)

Czechoslovak Mathematical Journal

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Let T ( q ) = k = 1 d ( k ) q k , | q | < 1 , where d ( k ) denotes the number of positive divisors of the natural number k . We present monotonicity properties of functions defined in terms of T . More specifically, we prove that H ( q ) = T ( q ) - log ( 1 - q ) log ( q ) is strictly increasing on ( 0 , 1 ) , while F ( q ) = 1 - q q H ( q ) is strictly decreasing on ( 0 , 1 ) . These results are then applied to obtain various inequalities, one of which states that the double inequality α q 1 - q + log ( 1 - q ) log ( q ) < T ( q ) < β q 1 - q + log ( 1 - q ) log ( q ) , 0 < q < 1 , holds with the best possible constant factors α = γ and β = 1 . Here, γ denotes Euler’s constant. This refines a result of Salem, who...

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there...

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let L ( H ) denote the algebra of operators on a complex infinite dimensional Hilbert space H . For A , B L ( H ) , the generalized derivation δ A , B and the elementary operator Δ A , B are defined by δ A , B ( X ) = A X - X B and Δ A , B ( X ) = A X B - X for all X L ( H ) . In this paper, we exhibit pairs ( A , B ) of operators such that the range-kernel orthogonality of δ A , B holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Δ A , B with respect to the wider class of unitarily invariant...

Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

Boban Karapetrović (2018)

Czechoslovak Mathematical Journal

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We show that if α > 1 , then the logarithmically weighted Bergman space A log α 2 is mapped by the Libera operator into the space A log α - 1 2 , while if α > 2 and 0 < ε α - 2 , then the Hilbert matrix operator H maps A log α 2 into A log α - 2 - ε 2 .We show that the Libera operator maps the logarithmically weighted Bloch space log α , α , into itself, while H maps log α into log α + 1 .In Pavlović’s paper (2016) it is shown that maps the logarithmically weighted Hardy-Bloch space log α 1 , α > 0 , into log α - 1 1 . We show that this result is sharp. We also show that H maps log α 1 , α 0 ,...

Uniform algebras and analytic multi­functions

Zbigniew Slodkowski (1983)

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 ( A ) . Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione λ log max g ( f - 1 ( λ ) ) è subarmonica su G e che l’applicazione λ g ( f - 1 ( λ ) ) è analitica nel senso di Oka.

Equilibrium states for interval maps: the potential - t log | D f |

Henk Bruin, Mike Todd (2009)

Annales scientifiques de l'École Normale Supérieure

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Let f : I I be a C 2 multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential φ t : x - t log | D f ( x ) | for t close to 1 , and also that the pressure function t P ( φ t ) is analytic on an appropriate interval near t = 1 .

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Eric Leichtnam, Xiang Tang, Alan Weinstein (2007)

Journal of the European Mathematical Society

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Let X be a complex manifold with strongly pseudoconvex boundary M . If ψ is a defining function for M , then log ψ is plurisubharmonic on a neighborhood of M in X , and the (real) 2-form σ = i ¯ ( log ψ ) is a symplectic structure on the complement of M in a neighborhood of M in X ; it blows up along M . The Poisson structure obtained by inverting σ extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure...

Multiplication operators on L ( L p ) and p -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 p < 2 , and T | X is an isomorphism for some reflexive subspace X of L p , then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.