Displaying similar documents to “On a sum involving the integral part function”

Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk (2020)

Commentationes Mathematicae Universitatis Carolinae

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Given a Tychonoff space X and an infinite cardinal κ , we prove that exponential κ -domination in X is equivalent to exponential κ -cofinality of C p ( X ) . On the other hand, exponential κ -cofinality of X is equivalent to exponential κ -domination in C p ( X ) . We show that every exponentially κ -cofinal space X has a κ + -small diagonal; besides, if X is κ -stable, then n w ( X ) κ . In particular, any compact exponentially κ -cofinal space has weight not exceeding κ . We also establish that any exponentially κ -cofinal...

On k -free numbers over Beatty sequences

Wei Zhang (2023)

Czechoslovak Mathematical Journal

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We consider k -free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number α > 1 of finite type τ < and any constant ε > 0 , we can show that 1 n x [ α n + β ] 𝒬 k 1 - x ζ ( k ) x k / ( 2 k - 1 ) + ε + x 1 - 1 / ( τ + 1 ) + ε , where 𝒬 k is the set of positive k -free integers and the implied constant depends only on α , ε , k and β . This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type 1 h H 1 n x n 𝒬 k e ( ϑ h n ) .

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 &gt; 1 , b &gt; 1 , c &gt; 0 , r &gt; 0 and s &gt; 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 ) &lt; 2 · 10 15 for each solution; when gcd ( a , b ) &gt; 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.

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 .

On an additive problem of unlike powers in short intervals

Qingqing Zhang (2022)

Czechoslovak Mathematical Journal

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We prove that almost all positive even integers n can be represented as p 2 2 + p 3 3 + p 4 4 + p 5 5 with | p k k - 1 4 N | N 1 - 1 / 54 + ε for 2 k 5 . As a consequence, we show that each sufficiently large odd integer N can be written as p 1 + p 2 2 + p 3 3 + p 4 4 + p 5 5 with | p k k - 1 5 N | N 1 - 1 / 54 + ε for 1 k 5 .

Complex series and connected sets

B. Jasek

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CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO Σ 1 , Σ 2 , Σ 3 , Σ 4 INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

On the exponential diophantine equation x y + y x = z z

Xiaoying Du (2017)

Czechoslovak Mathematical Journal

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For any positive integer D which is not a square, let ( u 1 , v 1 ) be the least positive integer solution of the Pell equation u 2 - D v 2 = 1 , and let h ( 4 D ) denote the class number of binary quadratic primitive forms of discriminant 4 D . If D satisfies 2 D and v 1 h ( 4 D ) 0 ( mod D ) , then D is called a singular number. In this paper, we prove that if ( x , y , z ) is a positive integer solution of the equation x y + y x = z z with 2 z , then maximum max { x , y , z } < 480000 and both x , y are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions...

Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space

Michael Gil&#039; (2023)

Czechoslovak Mathematical Journal

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We consider the equation d y ( t ) / d t = ( A + B ( t ) ) y ( t ) ( t 0 ) , where A is the generator of an analytic semigroup ( e A t ) t 0 on a Banach space 𝒳 , B ( t ) is a variable bounded operator in 𝒳 . It is assumed that the commutator K ( t ) = A B ( t ) - B ( t ) A has the following property: there is a linear operator S having a bounded left-inverse operator S l - 1 such that S e A t is integrable and the operator K ( t ) S l - 1 is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...