The square-free kernel of $x^{2^n} - a^{2^n}$

Paulo Ribenboim

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Displaying similar documents to “The square-free kernel of $x^{2^{n}} - a^{2^{n}}$ ”

On fixed point free involutions of $R^{1} \times S^{2}$

Gerhard X. Ritter (1976)

Colloquium Mathematicae

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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.

Consecutive square-free values of the type $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

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We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$ . We also establish an asymptotic formula for $1 \leq x, y, z \leq H$ such that $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$ are square-free. The method we used in this paper is due to Tolev.

Formally self-referential propositions for cut free classical analysis and related systems

G. Kreisel, G. Takeuti

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CONTENTSIntroduction............................................................................................................................................................................................................ 5 I. Results on self-referential propositions............................................................................................................................. 11 1. Definitions of some principal metamathematical notions......................................................................

A note on certain partial sum operators

Marek Bożejko, Gero Fendler (2006)

Banach Center Publications

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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of $L_{p}$ functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative $L_{p}$ spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

The range of non-linear natural polynomials cannot be context-free

Dömötör Pálvölgyi (2020)

Kybernetika

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Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i. e., $L = {f (n) ∣ n \in ℕ} \subseteq ℕ$ . We show that the base- $q$ representation of $L$ is a context-free language if and only if $f$ is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

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 $_{f} \subseteq G * H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_{f} {= p |}_{f} :_{f} \to G$ . A right inverse (section) $G \to_{f}$ 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 $k$ -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 $τ < \infty$ and any constant $ε > 0$ , we can show that $\sum_{\binom{1 \leq n \leq x}{[α n + β] \in 𝒬_{k}}} 1 - \frac{x}{ζ (k)} ≪ x^{k / (2 k - 1) + ε} + x^{1 - 1 / (τ + 1) + ε},$ 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 $\sum_{1 \leq h \leq H} \sum_{\binom{1 \leq n \leq x}{n \in 𝒬_{k}}} e (ϑ h n) .$

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u ₙ + u_{\infty}$ where uₙ and $u_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{α} + i S_{β}$ where $S_{α}$ and $S_{β}$ are free semicircular elements of variance α and β.

The Herz-Schur multiplier norm of sets satisfying the Leinert condition

Éric Ricard, Ana-Maria Stan (2011)

Colloquium Mathematicae

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It is well known that in a free group , one has $| | χ_{E} {| |}_{M_{c b} A ()} \leq 2$ , where E is the set of all the generators. We show that the (completely) bounded multiplier norm of any set satisfying the Leinert condition depends only on its cardinality. Consequently, based on a result of Wysoczański, we obtain a formula for $| | χ_{E} {| |}_{M_{c b} A ()}$ .

Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of $S_{\infty}$ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty}$ can be extended to a large free subgroup of $S_{\infty}$ , and, under Martin’s Axiom, any free subgroup of $S_{\infty}$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty}$ . Finally, if Gₙ are countable...

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Free locally convex spaces and $L$ -retracts

Rodrigo Hidalgo Linares, Oleg Okunev (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study the relation of $L$ -equivalence defined between Tychonoff spaces, that is, we study the topological isomorphisms of their respective free locally convex spaces. We introduce the concept of an $L$ -retract in a Tychonoff space in terms of the existence of a special kind of simultaneous extensions of continuous functions, explore the relation of this concept with the Dugundji extension theorem, and find some conditions that allow us to identify $L$ -retracts in various classes of topological...

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 $n^{f} \equiv 1 (mod p)$ 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 $\sum_{n \leq x} A (n) A (n + 1),$ and give an asymptotic formula by using properties of character...

Separable $ℵ_{k}$ -free modules with almost trivial dual

Daniel Herden, Héctor Gabriel Salazar Pedroza (2016)

Commentationes Mathematicae Universitatis Carolinae

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An $R$ -module $M$ has an almost trivial dual if there are no epimorphisms from $M$ to the free $R$ -module of countable infinite rank $R^{(ω)}$ . For every natural number $k > 1$ , we construct arbitrarily large separable $ℵ_{k}$ -free $R$ -modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

Displaying similar documents to “The square-free kernel of x 2 n - a 2 n ”

Displaying similar documents to “The square-free kernel of $x^{2^{n}} - a^{2^{n}}$ ”