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Generalized weighted quasi-arithmetic means
Aequationes mathematicae, 2010Let \(I\subseteq \mathbb R\) be an interval. A function \(M:\;I^2\to \mathbb R\) is called a mean on \(I^2\), if \[ \min (x,y)\leq M(x,y)\leq \max (x,y),\quad x,y\in I. \] The author considers means of the form \[ M_{f,g}(x,y)=(f+g)^{-1}(f(x)+g(y)) \] where \(f\) and \(g\) are real functions on \(I\), and studies conditions on \(f,g\), under which ...
Janusz Matkowski
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Limit properties of quasi-arithmetic means
Fuzzy Sets and Systems, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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An Orness Measure for Quasi-Arithmetic Means
IEEE Transactions on Fuzzy Systems, 2006In this paper, an orness measure to reflect the or-like degree of the quasi-arithmetic mean operator is proposed. With the generating function representation method, some properties of a quasi-arithmetic mean, associated with its orness measure, are analyzed.
Xinwang Liu
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On linear combinations of weighted quasi-arithmetic means
Aequationes mathematicae, 2005Let \(CM(I)\) denote the set of all continuous and strictly monotone real functions on the interval \(I\). A mean \(M\) on \(I\) is called a weighted quasi--arithmetic mean if there exists \(\phi \in CM(I)\) such that \[ M(x,y)=\phi^{-1}(\lambda\phi(x)+(1-\lambda)\phi(y))=:A_{\phi}(x,y;\lambda) \qquad (x,y \in I).
Daróczy, Zoltán, Hajdu, Gabriella
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Parameterized defuzzification with continuous weighted quasi-arithmetic means – An extension☆
Information Sciences, 2009zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Xinwang Liu
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1988
The power means n [r] (a;w), reR, defined in the previous chapter can be looked at in the following way; for each reR define a function φ as follows: Φ(x) = xr, r ≠ 0, Φ(x) = log x, r = 0, then $$M_n^{[r]}(\underline a ;\underline w ) = {\phi ^{ - 1}}\quad (\frac{1}{{{w_n}}}\sum\limits_{i = 1}^n {{w_i}\;\phi ({a_i})} ).$$ (1) This suggests ...
P. S. Bullen +2 more
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The power means n [r] (a;w), reR, defined in the previous chapter can be looked at in the following way; for each reR define a function φ as follows: Φ(x) = xr, r ≠ 0, Φ(x) = log x, r = 0, then $$M_n^{[r]}(\underline a ;\underline w ) = {\phi ^{ - 1}}\quad (\frac{1}{{{w_n}}}\sum\limits_{i = 1}^n {{w_i}\;\phi ({a_i})} ).$$ (1) This suggests ...
P. S. Bullen +2 more
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Invariance equation for generalized quasi-arithmetic means
Aequationes mathematicae, 2009In this paper, the invariance equation $$(\varphi_{1} + \varphi_{2})^{-1} (\varphi_{1}(x) + \varphi_{2}(y)) + (\psi_{1} + \psi_{2})^{-1}(\psi_{1}(x) + \psi_{2}(y)) = x + y$$ is solved under four times continuous differentiability of the unknown functions φ1, φ2, ψ1, ψ2.
Szabolcs Baják, Zsolt Páles
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The Matkowski–Sutő problem for weighted quasi-arithmetic means
Acta Mathematica Hungarica, 2003Let \(I\subset\mathbb{R}\) be a non-void open interval and let \(\mathcal{CM}(I)\) denote the class of all continuous and strictly monotone real-valued functions defined on the interval \(I\). A function \(M:I\times I \to I\) is called a weighted quasi-arithmetic mean on \(I\) if there exist a number ...
Daróczy, Z., Páles, Zs.
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Invariant and complementary quasi-arithmetic means
Aequationes Mathematicae, 1999If \(I\) is a proper (non-singleton) real interval and \(M\) and \(N\) are continuous, both map \(I^{2}\) into \(I\), and both \(M(x,y)\) and \(N(x,y)\) lie between \(\min(x,y)\) and \(\max(x,y),\) one of them always strictly between if \(x\neq y\) (that is, both are means and one of them is a strict mean), then it is easy to see that there exists a ...
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