Results 271 to 280 of about 227,128 (319)

An Entropy Proof of the Arithmetic Mean–Geometric Mean Inequality

The American mathematical monthly, 2020
Many proofs are known for the inequality between the arithmetic mean and the geometric mean. This note gives a new derivation, interpreting the means as final and initial values of entropy, and the inequality as the second law of thermodynamics.
Cole Graham, T. Tokieda, V. Ponomarenko
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The Arithmetic-Geometric Mean of Gauss

1997
This 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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Notes on the arithmetic–geometric mean inequality

Aequationes mathematicae
Starting with the scalar inequality \[ |(ab)^{1/2}x+(ab)^{-1/2}y|\le \frac{|ax + b^{-1}y|+|bx + a^{-1}y|}{2}, \] where \(a, b\) are positive real numbers and \(x, y\) are complex numbers, the authors present matrix versions of it and some generalizations.
Al-Natoor, Ahmad, Hirzallah, Omar, Kittaneh, Fuad   +2 more
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The Arithmetic, Geometric and Harmonic Means

1988
This chapter is devoted to the properties and inequalities of the classical arithmetic, geometric and harmonic means. In particular the basic inequality between these means, the Geometric Mean-Arithmetic Mean Inequality, is discussed at length with many proofs being given.
P. S. Bullen, D. S. Mitrinović, P. M. Vasić   +2 more
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A noncontinuous generalization of the arithmetic–geometric mean

Applied Mathematics and Computation, 2001
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ekárt, Anikó, Németh, S. Z.
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Geometric-Mean Policy Optimization

arXiv.org
Group 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, Yue Liu, Junpeng Liu, Jingye Chen, Xun Wu, Y. Hao, Tengchao Lv, Shaohan Huang, Lei Cui, Qixiang Ye, Fang Wan, Furu Wei   +11 more
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Arithmetic-geometric means for electrical energies

Proceedings of the IEEE, 1980
The paper indicates an important arithmetic-geometric relation for the behavior of linear passive reciprocal n-ports. The active energies dissipated in the unit time interval in any linear passive time invariant reciprocal resistive system satisfy the inequalities P 1 + P 2 ≥ 2P 0 √P 1 P 2 ≥ P 0 where P 1 = (V 1 , I 1 ), P 2 = (V 2 , I 2 ), P 0 = (V 1 ,
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Arithmetic–Geometric Mean and Related Submajorisation and Norm Inequalities for $$\tau $$-Measurable Operators: Part II

Integral equations and operator theory, 2020
P. Dodds, T. Dodds, F. Sukochev, D. Zanin   +3 more
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More Matrix Forms of the Arithmetic-Geometric Mean Inequality

SIAM Journal on Matrix Analysis and Applications, 1993
The authors prove the following arithmetic-geometric mean inequality: \(2||| A^* XB||| \leq ||| AA^* X+XBB^*|||\) for arbitrary \(n\times n\) matrices \(A\), \(B\), \(X\). They also show that the real function \(f(p):=||| A^{1+p} XB^{1-p}+A^{1-p} XB^{1+p}|||\), \(A,B\geq 0\) is convex on \([-1,1]\) and takes its minimum at \(p=0\), where \(|||\cdot|||\)
Bhatia, Rajendra, Davis, Chandler
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The arithmetic-geometric mean and the elliptic mean error

Acta Geodaetica et Geophysica Hungarica, 2003
Without any special term, the mathematical definition of a measuring index for reliability of geodetic point was given by Lajos Homorodi. For this index, the term of the elliptic mean error was proposed by the author of the present paper and it was shown that the elliptic mean error is beneficial to the being for characterizing the reliability of ...
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