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Displaying similar documents to “GCD sums from Poisson integrals and systems of dilated functions”

Prolongation of Poisson 2 -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

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In this paper, M denotes a smooth manifold of dimension n , A a Weil algebra and M A the associated Weil bundle. When ( M , ω M ) is a Poisson manifold with 2 -form ω M , we construct the 2 -Poisson form ω M A A , prolongation on M A of the 2 -Poisson form ω M . We give a necessary and sufficient condition for that M A be an A -Poisson manifold.

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form W F , d ( a ) = x F ψ ( x d - a x ) , where F is a finite field, ψ: F → ℂ is the canonical additive character, g c d ( d , | F × | ) = 1 , and a F × . If we fix F and d, and examine the values of W F , d ( a ) as a runs through F × , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo | F × | ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

Repdigits in the base b as sums of four balancing numbers

Refik Keskin, Faticko Erduvan (2021)

Mathematica Bohemica

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The sequence of balancing numbers ( B n ) is defined by the recurrence relation B n = 6 B n - 1 - B n - 2 for n 2 with initial conditions B 0 = 0 and B 1 = 1 . B n is called the n th balancing number. In this paper, we find all repdigits in the base b , which are sums of four balancing numbers. As a result of our theorem, we state that if B n is repdigit in the base b and has at least two digits, then ( n , b ) = ( 2 , 5 ) , ( 3 , 6 ) . Namely, B 2 = 6 = ( 11 ) 5 and B 3 = 35 = ( 55 ) 6 .

The number of solutions to the generalized Pillai equation ± r a x ± s b y = c .

Reese Scott, Robert Styer (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider N , the number of solutions ( x , y , u , v ) to the equation ( - 1 ) u r a x + ( - 1 ) v s b y = c in nonnegative integers x , y and integers u , v { 0 , 1 } , for given integers a > 1 , b > 1 , c > 0 , r > 0 and s > 0 . When gcd ( r a , s b ) = 1 , we show that N 3 except for a finite number of cases all of which satisfy max ( a , b , r , s , x , y ) < 2 · 10 15 for each solution; when gcd ( a , b ) > 1 , we show that N 3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving N = 3 solutions.

( 0 , 1 ) -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let m and n be positive integers, and let R = ( r 1 , ... , r m ) and S = ( s 1 , ... , s n ) be nonnegative integral vectors. Let A ( R , S ) be the set of all m × n ( 0 , 1 ) -matrices with row sum vector R and column vector S . Let R and S be nonincreasing, and let F ( R ) be the m × n ( 0 , 1 ) -matrix, where for each i , the i th row of F ( R , S ) consists of r i 1’s followed by ( n - r i ) 0’s. Let A A ( R , S ) . The discrepancy of A, disc ( A ) , is the number of positions in which F ( R ) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over...

On sums and products in a field

Guang-Liang Zhou, Zhi-Wei Sun (2022)

Czechoslovak Mathematical Journal

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We study sums and products in a field. Let F be a field with ch ( F ) 2 , where ch ( F ) is the characteristic of F . For any integer k 4 , we show that any x F can be written as a 1 + + a k with a 1 , , a k F and a 1 a k = 1 , and that for any α F { 0 } we can write every x F as a 1 a k with a 1 , , a k F and a 1 + + a k = α . We also prove that for any x F and k { 2 , 3 , } there are a 1 , , a 2 k F such that a 1 + + a 2 k = x = a 1 a 2 k .

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...

On perfect powers in k -generalized Pell sequence

Zafer Şiar, Refik Keskin, Elif Segah Öztaş (2023)

Mathematica Bohemica

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Let k 2 and let ( P n ( k ) ) n 2 - k be the k -generalized Pell sequence defined by P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + + P n - k ( k ) for n 2 with initial conditions P - ( k - 2 ) ( k ) = P - ( k - 3 ) ( k ) = = P - 1 ( k ) = P 0 ( k ) = 0 , P 1 ( k ) = 1 . In this study, we handle the equation P n ( k ) = y m in positive integers n , m , y , k such that k , y 2 , and give an upper bound on n . Also, we will show that the equation P n ( k ) = y m with 2 y 1000 has only one solution given by P 7 ( 2 ) = 13 2 .

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair ( 𝒳 , 𝒜 ) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ . The space 𝒜 is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p : 𝒜 𝒳 . The space 𝒜 is equipped with a closed 2 -form, possibly degenerate, and the space 𝒳 has a Poisson structure. The map p is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that real functions on which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower C 1 -functions, or of two strongly paraconvex functions) coincide with semismooth functions on (i.e. those locally Lipschitz functions on for which f + ' ( x ) = lim t x + f + ' ( t ) and f - ' ( x ) = lim t x - f - ' ( t ) for each x ). Further, for each modulus ω , we characterize the class D S C ω of functions on which can be written as f = g - h , where g and h are semiconvex with modulus C ω (for some C > 0 ) using a new...

Waring's number for large subgroups of ℤ*ₚ*

Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer (2014)

Acta Arithmetica

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Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain t = p 22 / 39 + ϵ , t = p 15 / 29 + ϵ and for s ≥ 6, t = p ( 9 s + 45 ) / ( 29 s + 33 ) + ϵ . For s ≥ 24 further improvements are made, such as t 32 = p 5 / 16 + ϵ and t 128 = p 1 / 4 .

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

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For an integer k 2 , let ( n ) n be the k - generalized Pell sequence which starts with 0 , ... , 0 , 1 ( k terms) and each term afterwards is given by the linear recurrence n = 2 n - 1 + n - 2 + + n - k . In this paper, we find all k -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ( P n ( 2 ) ) n . ...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .