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Displaying similar documents to “Consecutive square-free values of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1

On the r -free values of the polynomial x 2 + y 2 + z 2 + k

Gongrui Chen, Wenxiao Wang (2023)

Czechoslovak Mathematical Journal

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Let k be a fixed integer. We study the asymptotic formula of R ( H , r , k ) , which is the number of positive integer solutions 1 x , y , z H such that the polynomial x 2 + y 2 + z 2 + k is r -free. We obtained the asymptotic formula of R ( H , r , k ) for all r 2 . Our result is new even in the case r = 2 . We proved that R ( H , 2 , k ) = c k H 3 + O ( H 9 / 4 + ε ) , where c k > 0 is a constant depending on k . This improves upon the error term O ( H 7 / 3 + ε ) obtained by G.-L. Zhou, Y. Ding (2022).

Pairs of square-free values of the type n 2 + 1 , n 2 + 2

Stoyan Dimitrov (2021)

Czechoslovak Mathematical Journal

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We show that there exist infinitely many consecutive square-free numbers of the form n 2 + 1 , n 2 + 2 . We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number.

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 ) .

Equalizers and coactions of groups

Martin Arkowitz, Mauricio Gutierrez (2002)

Fundamenta Mathematicae

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If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group f G * H be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism p f = p | f : f G . A right inverse (section) G f of p f is called a coaction on G. In this paper we study f and the sections of p f . We consider the following topics: the structure of f as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

On the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

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Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

On the distribution of consecutive square-free primitive roots modulo p

Huaning Liu, Hui Dong (2015)

Czechoslovak Mathematical Journal

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A positive integer n is called a square-free number if it is not divisible by a perfect square except 1 . Let p be an odd prime. For n with ( n , p ) = 1 , the smallest positive integer f such that n f 1 ( mod p ) is called the exponent of n modulo p . If the exponent of n modulo p is p - 1 , then n is called a primitive root mod p . Let A ( n ) be the characteristic function of the square-free primitive roots modulo p . In this paper we study the distribution n x A ( n ) A ( n + 1 ) , and give an asymptotic formula by using properties of character...

Coleff-Herrera currents, duality, and noetherian operators

Mats Andersson (2011)

Bulletin de la Société Mathématique de France

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Let be a coherent subsheaf of a locally free sheaf 𝒪 ( E 0 ) and suppose that = 𝒪 ( E 0 ) / has pure codimension. Starting with a residue current R obtained from a locally free resolution of we construct a vector-valued Coleff-Herrera current μ with support on the variety associated to such that φ is in if and only if μ φ = 0 . Such a current μ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

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...