Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski

Displaying similar documents to “Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem”

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

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We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f (M (x, y)) = A_{φ} (f (x), f (y))$ and $f (A_{φ} (x, y)) = M (f (x), f (y))$ are the constant ones.

Iterated quasi-arithmetic mean-type mappings

Paweł Pasteczka (2016)

Colloquium Mathematicae

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We work with a fixed N-tuple of quasi-arithmetic means $M ₁, . . ., M_{N}$ generated by an N-tuple of continuous monotone functions $f ₁, . . ., f_{N} : I \to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^{N} ∋ b \mapsto (M ₁ (b), . . ., M_{N} (b))$ tend pointwise to a mapping having values on the diagonal of $I^{N}$ . Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M ₁, . . ., M_{N}$ taken...

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

R. P. Pakshirajan (1963)

Annales Polonici Mathematici

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On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

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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 $f$ .

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. ...

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 $\hat{\deg} ({\bar{D}}^{2}) = \hat{vol} (\bar{D})$ .

On the least almost-prime in arithmetic progressions

Liuying Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒫_{2}$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$ , $q$ such that $(a, q) = 1$ , let $𝒫_{2} (q, a)$ denote the least $𝒫_{2}$ in the arithmetic progression ${n q + a}_{n = 1}^{\infty}$ . It is proved that for sufficiently large $q$ , we have $𝒫_{2} (q, a) ≪ q^{1.825} .$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained $𝒫_{2} (q, a) ≪ q^{1.8345} .$

A problem of Rankin on sets without geometric progressions

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

Acta Arithmetica

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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...

Order convexity and concavity of Lorentz spaces $Λ_{p, w}$ , 0 < p < ∞

Anna Kamińska, Lech Maligranda (2004)

Studia Mathematica

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We study order convexity and concavity of quasi-Banach Lorentz spaces $Λ_{p, w}$ , where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that $Λ_{p, w}$ contains an order isomorphic copy of $l^{p}$ . We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for $Λ_{p, w}$ . We conclude with a characterization of the type and cotype of $Λ_{p, w}$ in the case when $Λ_{p, w}$ is a normable space.

A functional equation involving f and $f^{- 1}$ .

J. S. Ratti, Y. -F. Lin (1990)

Colloquium Mathematicae

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Coprimality of integers in Piatetski-Shapiro sequences

Watcharapon Pimsert, Teerapat Srichan, Pinthira Tangsupphathawat (2023)

Czechoslovak Mathematical Journal

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We use the estimation of the number of integers $n$ such that $⌊ n^{c} ⌋$ belongs to an arithmetic progression to study the coprimality of integers in $ℕ^{c} = {⌊ n^{c} ⌋}_{n \in ℕ}$ , $c > 1$ , $c \notin ℕ$ .

Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝔐$ and $ℜ$ be algebras of subsets of a set $Ω$ with $𝔐 \subset ℜ$ , and denote by $E (μ)$ the set of all quasi-measure extensions of a given quasi-measure $μ$ on $𝔐$ to $ℜ$ . We give some criteria for order boundedness of $E (μ)$ in $b a (ℜ)$ , in the general case as well as for atomic $μ$ . Order boundedness implies weak compactness of $E (μ)$ . We show that the converse implication holds under some assumptions on $𝔐$ , $ℜ$ and $μ$ or $μ$ alone, but not in general.