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On Means That are Both Quasi-Arithmetic and Conjugate Arithmetic
Acta Mathematica Hungarica, 2001The authors determine all means of two variables that are simultaneously of the form \[ \psi^{-1}\biggl({\psi(x)+\psi(y)\over 2}\biggr)\quad\text{and}\quad \varphi^{-1}(\varphi(x)+\varphi(y)-\varphi\Bigl({x+y\over 2}\Bigl)). \] The functions \(\psi\) and \(\varphi\) are strictly monotonic, continuous, defined on an open real interval, and one of them ...
Daróczy, Z., Páles, Zs.
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The Arithmetic Teacher, 1957
There are so many meanings in arithmetic that the only difficulty in writing on this subject is one of selection. The following ideas are miscellaneous rather than organized.
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There are so many meanings in arithmetic that the only difficulty in writing on this subject is one of selection. The following ideas are miscellaneous rather than organized.
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Arithmetic Facts without Meaning
Cortex, 1997The paper presents a single-case study of patient J.G. showing severe calculation problems (and also agraphia, finger agnosia, right-left disorientation and apraxia) after the surgery of a left parietal tumor. Although the patient completely lost conceptual knowledge of arithmetic, she preserved part of memorised fact knowledge (multiplications and ...
M, Delazer, T, Benke
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Arithmetic mean as a linear combination of two quasi-arithmetic means
Publicationes Mathematicae Debrecen, 2002Under twice continuous differentiability assumptions on the generator functions \(f\) and \(g\), the authors determine all two variable quasi-arithmetic means \(M^{[f]}\), \(M^{[g]}\) and all real numbers \(\lambda\) and \(\mu\) such that \[ \lambda M^{[f]}+\mu M^{[g]}=A, \] where \(A\) denotes the two variable arithmetic mean.
Głazowska, Dorota +2 more
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The Arithmetic-Geometric Mean of Gauss
1997This paper is an expository account of the arithmetic-geometric mean M(a,b) of two numbers a,b. For \(a,b>0\) define \(a_ 0=a\), \(b_ 0=b\) and \(a_{n+1}=(a_ n+b_ n)/2,\quad b_{n+1}=(a_ nb_ n)^{1/2},\quad n=0,1,2,\ldots.\) It follows by elementary methods that the two sequences \(a_ n\), \(b_ n\) have a common limit M(a,b).
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Meanings of savarṇana in Indian Arithmetic*
Ganita Bharati, 2023Summary: In Indian mathematical texts the term savarṇana ``reduction to the same color'' is usually found in the context of calculation for fractions. A number of explanations for the term have been offered in previous studies, but they slightly differ from each other. The Triśatī{b}hāṣya is an anonymous commentary on Śrīdhara's Triśatī. In the present
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THE MEAN-VALUES OF ARITHMETICAL FUNCTIONS
The Quarterly Journal of Mathematics, 1949Es wird zunächst folgender Satz bewiesen: Es seien \(\{a_n\}\), \(\{A_n\}\) zwei Folgen, verknüpft durch \[ A_n = \sum_{m\mid n} a_m,\quad a_n = \sum_{m\mid n} \mu(n) A_m, \] dann gilt \[ \lim_{N\to\infty} N^{-1} \sum_{n=1}^N A_n \rightarrow \lim_{N\to\infty} N^{-1} \sum_{n=1}^N a_nn^{-1}, \qquad (N\to\infty), \] wenn es eine positive nichtabnehmende ...
Atkinson, F. V., Lord Cherwell
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An Analysis of Meaning in Arithmetic. II
The Elementary School Journal, 1949Thinking educators recognize that children of all ages need to have arithmetic related to their own experiences. Such a relation not only enriches living but lays an essential background for understanding arithmetic. It is only in this way that children may acquire the necessary understandings for arithmetical processes in a social background.12 ...
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A REMARK ON THE ARITHMETIC MEAN OF AN INFINITE SEQUENCE
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2002We discuss several aspects of the generalization of the arithmetic mean for infinite sequences. It is shown that no operator on infinite sequences can posses all properties of the classical arithmetic mean. Some alternative operators are introduced and their properties compared.
L. González, E. Muel, Radko Mesiar
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Geometric-Mean Policy Optimization
arXiv.orgGroup Relative Policy Optimization (GRPO) has significantly enhanced the reasoning capability of large language models by optimizing the arithmetic mean of token-level rewards.
Yuzhong Zhao +11 more
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