On embedding models of arithmetic of cardinality ℵ₁ into reduced powers

Juliette Kennedy; Saharon Shelah

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Displaying similar documents to “On embedding models of arithmetic of cardinality ℵ₁ into reduced powers”

Some properties of the class of arithmetic functions $T_{r} (N)$

R. P. Pakshirajan (1963)

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On a certain class of arithmetic functions

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A homothetic arithmetic function of ratio $K$ is a function $f : ℕ \to R$ such that $f (K n) = f (n)$ for every $n \in ℕ$ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f (ℕ)$ in terms of the period and the ratio of $f$ .

$P_{λ}$ -sets and skeletal mappings

Aleksander Błaszczyk, Anna Brzeska (2013)

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We prove that if the topology on the set Seq of all finite sequences of natural numbers is determined by $P_{λ}$ -filters and λ ≤ , then Seq is a $P_{λ}$ -set in its Čech-Stone compactification. This improves some results of Simon and of Juhász and Szymański. As a corollary we obtain a generalization of a result of Burke concerning skeletal maps and we partially answer a question of his.

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We present principles for guessing clubs in the generalized club filter on $_{κ} λ$ . These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ⁺-Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of ${[λ]}^{ℵ ₀}$ .

Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper, we give a numerical characterization of nef arithmetic $ℝ$ -Cartier divisors of $C^{0}$ -type on an arithmetic surface. Namely an arithmetic $ℝ$ -Cartier divisor $\bar{D}$ of $C^{0}$ -type is nef if and only if $\bar{D}$ is pseudo-effective and $\hat{\deg} ({\bar{D}}^{2}) = \hat{vol} (\bar{D})$ .

Counting models of set theory

Ali Enayat (2002)

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Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD,...

On the equation $n (n + d) \dots (n + (i ₀ - 1) d) (n + (i ₀ + 1) d) \dots (n + (k - 1) d) = y^{l}$ with 0 < i₀ < k-1

N. Saradha, T. N. Shorey (2007)

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On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

Filter factors of truncated TLS regularization with multiple observations

Iveta Hnětynková, Martin Plešinger, Jana Žáková (2017)

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The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems $A x \approx b$ were analyzed by Fierro, Golub, Hansen, and O’Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of $A$ applied to $b$ . This paper focuses...

Characterizing the powerset by a complete (Scott) sentence

Ioannis Souldatos (2013)

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This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence $ϕ_{ℳ}$ if $ϕ_{ℳ}$ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $ℵ_{β}$ is characterized by a Scott sentence, then $2^{ℵ_{β + β ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 <...

A note on the article by F. Luca “On the system of Diophantine equations $a ² + b ² = {(m ² + 1)}^{r}$ and $a^{x} + b^{y} = {(m ² + 1)}^{z}$ ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

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Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A + B \sqrt (- 1) = {(m + \sqrt (- 1))}^{r}$ . We prove that the Diophantine equation ${| A |}^{x} + {| B |}^{y} = {(m ² + 1)}^{z}$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r > 10^{74}$ or $m > 10^{34}$ . Our result is an explicit refinement of a theorem due to F. Luca.

Finiteness results for Diophantine triples with repdigit values

Attila Bérczes, Florian Luca, István Pink, Volker Ziegler (2016)

Acta Arithmetica

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Let g ≥ 2 be an integer and $_{g} \subset ℕ$ be the set of repdigits in base g. Let $_{g}$ be the set of Diophantine triples with values in $_{g}$ ; that is, $_{g}$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set $_{g}$ . We prove effective finiteness results for the set $_{g}$ .

On systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

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We consider systems of equations of the form $x_{i} + x_{j} = x_{k}$ and $x_{i} \cdot x_{j} = x_{k}$ , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

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We study the Diophantine equations ${(k!)}^{n} - k^{n} = {(n!)}^{k} - n^{k}$ and ${(k!)}^{n} + k^{n} = {(n!)}^{k} + n^{k},$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k = n$ or $(k, n) = (1, 2), (2, 1)$ and that the second one holds if and only if $k = n$ .

Products of topological spaces and families of filters

Paolo Lipparini (2023)

Commentationes Mathematicae Universitatis Carolinae

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We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. We prove that a product is Lindelöf if and only if all subproducts by $\leq ω_{1}$ factors are Lindelöf. Parallel results are obtained for final $ω_{n}$ -compactness, $[λ, μ]$ -compactness, the Menger and the Rothberger properties.

Existentially closed II₁ factors

Ilijas Farah, Isaac Goldbring, Bradd Hart, David Sherman (2016)

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We examine the properties of existentially closed ( $^{ω}$ -embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ( $^{ω}$ -embeddable) II₁ factor is approximately inner to prove that Th() is not model-complete. We also show that Th() is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th().

Higher order spreading models

S. A. Argyros, V. Kanellopoulos, K. Tyros (2013)

Fundamenta Mathematicae

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We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences ${(x_{s})}_{s \in ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy ${(ℳ_{ξ} (X))}_{ξ < ω ₁}$ . Each $ℳ_{ξ} (X)$ contains all spreading models generated by ℱ-sequences ${(x_{s})}_{s \in ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.

Stein’s method in high dimensions with applications

Adrian Röllin (2013)

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Let $h$ be a three times partially differentiable function on $ℝ^{n}$ , let $X = (X_{1}, ..., X_{n})$ be a collection of real-valued random variables and let $Z = (Z_{1}, ..., Z_{n})$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $𝔼 h (X) - 𝔼 h (Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n \to \infty$ . In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic...

A structure theorem for sets of small popular doubling

Przemysław Mazur (2015)

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We prove that every set A ⊂ ℤ satisfying $\sum_{x} m i n (1_{A} * 1_{A} (x), t) \leq (2 + δ) t | A |$ for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ (| ℕ ∖ (A + A) | \geq k) = Θ (2^{- k / 2})$ .

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

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A geometric progression of length k and integer ratio is a set of numbers of the form $a, a r, . . ., a r^{k - 1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${(a_{i})}_{i = 1}^{\infty}$ of positive real numbers with a₁ = 1 such that the set $G^{(k)} = ⋃_{i = 1}^{\infty} (a_{2 i}, a_{2 i - 1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...

On the diophantine equation $x^{y} - y^{x} = c^{z}$

Zhongfeng Zhang, Jiagui Luo, Pingzhi Yuan (2012)

Colloquium Mathematicae

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Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation $x^{y} - y^{x} = c^{z}$ with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either $(x, y, z) = (c^{k} + 1, 1, k)$ , k ≥ 1 or $2 \leq x < y \leq m a x 1.5 \times 10^{10}, c$ if c ≥ 2.

Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $F_{n}$ be the $n$ -th Fibonacci number. Put $ϕ = \frac{1 + \sqrt{5}}{2}$ . We prove that the following inequalities hold for any real $α$ : 1) ${inf}_{n \in ℕ} | | F_{n} α | | \leq \frac{ϕ - 1}{ϕ + 2}$ , 2) ${lim inf}_{n \to \infty} | | F_{n} α | | \leq \frac{1}{5}$ , 3) ${lim inf}_{n \to \infty} | | ϕ^{n} α | | \leq \frac{1}{5}$ . These results are the best possible.