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Displaying similar documents to “Exponential sums with coefficients 0 or 1 and concentrated L p norms”

On microlocal analyticity of solutions of first-order nonlinear PDE

Shif Berhanu (2009)

Annales de l’institut Fourier

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We study the microlocal analyticity of solutions u of the nonlinear equation u t = f ( x , t , u , u x ) where f ( x , t , ζ 0 , ζ ) is complex-valued, real analytic in all its arguments and holomorphic in ( ζ 0 , ζ ) . We show that if the function u is a C 2 solution, σ Char L u and 1 i σ ( [ L u , L u ¯ ] ) < 0 or if u is a C 3 solution, σ Char L u , σ ( [ L u , L u ¯ ] ) = 0 , and σ ( [ L u , [ L u , L u ¯ ] ] ) 0 , then σ W F a u . Here W F a u denotes the analytic wave-front set of u and Char L u is the characteristic set of the linearized operator. When m = 1 , we prove a more general result involving the repeated brackets of L u and L u ¯ of any...

A remark on a modified Szász-Mirakjan operator

Guanzhen Zhou, Songping Zhou (1999)

Colloquium Mathematicae

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We prove that, for a sequence of positive numbers δ(n), if n 1 / 2 δ ( n ) ¬ as n , to guarantee that the modified Szász-Mirakjan operators S n , δ ( f , x ) converge to f(x) at every point, f must be identically zero.

On the linear independence of p -adic L -functions modulo p

Bruno Anglès, Gabriele Ranieri (2010)

Annales de l’institut Fourier

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Let p 3 be a prime. Let n such that n 1 , let χ 1 , ... , χ n be characters of conductor d not divided by p and let ω be the Teichmüller character. For all i between 1 and n , for all j between 0 and ( p - 3 ) / 2 , set θ i , j = χ i ω 2 j + 1 if χ i is odd ; χ i ω 2 j if χ i is even . Let K = p ( χ 1 , ... , χ n ) and let π be a prime of the valuation ring 𝒪 K of K . For all i , j let f ( T , θ i , j ) be the Iwasawa series associated to θ i , j and f ( T , θ i , j ) ¯ its reduction modulo ( π ) . Finally let 𝔽 p ¯ be an algebraic closure of 𝔽 p . Our main result is that if the characters χ i are all distinct modulo ( π ) , then 1 and the series...

The summatory function of q -additive functions on pseudo-polynomial sequences

Manfred G. Madritsch (2012)

Journal de Théorie des Nombres de Bordeaux

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The present paper deals with the summatory function of functions acting on the digits of an q -ary expansion. In particular let n be a positive integer, then we call n = r = 0 d r ( n ) q r with d r ( n ) { 0 , ... , q - 1 } its q -ary expansion. We call a function f strictly q -additive, if for a given value, it acts only on the digits of its representation, i.e., f ( n ) = r = 0 f d r ( n ) . Let p ( x ) = α 0 x β 0 + + α d x β d with α 0 , α 1 , ... , α d , , α 0 > 0 , β 0 > > β d 1 and at least one β i . Then we call p a pseudo-polynomial. ...

A system of simultaneous congruences arising from trinomial exponential sums

Todd Cochrane, Jeremy Coffelt, Christopher Pinner (2006)

Journal de Théorie des Nombres de Bordeaux

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For a prime p and positive integers < k < h < p with d = ( h , k , , p - 1 ) , we show that M , the number of simultaneous solutions x , y , z , w in p * to x h + y h = z h + w h , x k + y k = z k + w k , x + y = z + w , satisfies M 3 d 2 ( p - 1 ) 2 + 25 h k ( p - 1 ) . When h k = o ( p d 2 ) we obtain a precise asymptotic count on M . This leads to the new twisted exponential sum bound x = 1 p - 1 χ ( x ) e 2 π i f ( x ) / p 3 1 4 d 1 2 p 7 8 + 5 h k 1 4 p 5 8 , for trinomials f = a x h + b x k + c x , and to results on the average size of such sums.