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

G. Kreisel; G. Takeuti

Displaying similar documents to “Formally self-referential propositions for cut free classical analysis and related systems”

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

Gerhard X. Ritter (1976)

Colloquium Mathematicae

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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.

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.

On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

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...

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...

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 $\sum_{1 \leq h \leq H} \sum_{\binom{1 \leq n \leq x}{n \in 𝒬_{k}}} e (ϑ h n) .$

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 $1 \leq x, y, z \leq H$ 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 $r \geq 2$ . 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).

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .

The $ℒ_{n}^{m}$ -propositional calculus

Carlos Gallardo, Alicia Ziliani (2015)

Mathematica Bohemica

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T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the $ℒ_{n}^{m}$ -propositional calculus, denoted by $ℓ_{n}^{m}$ , is introduced in terms of the binary connectives $\to$ (implication), $↠$ (standard implication), $\land$ (conjunction), $\lor$ (disjunction) and the unary ones $f$ (negation) and $D_{i}$ , $1 \leq i \leq n - 1$ (generalized Moisil operators). It is proved that $ℓ_{n}^{m}$ belongs to the class of standard systems...

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...

Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space

Michael Gil' (2024)

Czechoslovak Mathematical Journal

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Let $A$ be a bounded linear operator in a complex separable Hilbert space $ℋ$ , and $S$ be a selfadjoint operator in $ℋ$ . Assuming that $A - S$ belongs to the Schatten-von Neumann ideal $𝒮_{p}$ $(p > 1),$ we derive a bound for $\sum_{k} {| R λ_{k} (A) - λ_{k} (S) |}^{p}$ , where $λ_{k} (A)$ $(k = 1, 2, \dots)$ are the eigenvalues of $A$ . Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p = 2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues...