• Title, Summary, Keyword: arithmetic mean-geometric mean inequality

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Weighted Geometric Means of Positive Operators

  • Izumino, Saichi;Nakamura, Noboru
    • Kyungpook Mathematical Journal
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    • v.50 no.2
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    • pp.213-228
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    • 2010
  • A weighted version of the geometric mean of k ($\geq\;3$) positive invertible operators is given. For operators $A_1,{\ldots},A_k$ and for nonnegative numbers ${\alpha}_1,\ldots,{\alpha}_k$ such that $\sum_\limits_{i=1}^k\;\alpha_i=1$, we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to $A_1^{\alpha_1}{\cdots}A_k^{{\alpha}_k}$ if $A_1,{\ldots},A_k$ commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

A study on expression of students in the process of constructing average concept as mathematical knowledge (수학적 지식으로서의 평균 개념 구성 과정에서 나타난 학생들의 표현에 관한 연구)

  • Lee, Dong Gun
    • The Mathematical Education
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    • v.57 no.3
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    • pp.311-328
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    • 2018
  • In school mathematics, the concept of an average is not a concept that is limited to a unit of statistics. In particular, high school students will learn about arithmetic mean and geometric mean in the process of learning absolute inequality. In calculus learning, the concept of average is involved when learning the concept of average speed. The arithmetic mean is the same as the procedure used when students mean the test scores. However, the procedure for obtaining the geometric mean differs from the procedure for the arithmetic mean. In addition, if the arithmetic mean and the geometric mean are the discrete quantity, then the mean rate of change or the average speed is different in that it considers continuous quantities. The average concept that students learn in school mathematics differs in the quantitative nature of procedures and objects. Nevertheless, it is not uncommon to find out how students construct various mathematical concepts into mathematical knowledge. This study focuses on this point and conducted the interviews of the students(three) in the second grade of high school. And the expression of students in the process of average concept formation in arithmetic mean, geometric mean, average speed. This study can be meaningful because it suggests practical examples to students about the assertion that various scholars should experience various properties possessed by the average. It is also meaningful that students are able to think about how to construct the mean conceptual properties inherent in terms such as geometric mean and mean speed in arithmetic mean concept through interview data.

Geometric Means of Positive Operators

  • Nakamura, Noboru
    • Kyungpook Mathematical Journal
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    • v.49 no.1
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    • pp.167-181
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    • 2009
  • Based on Ricatti equation $XA^{-1}X=B$ for two (positive invertible) operators A and B which has the geometric mean $A{\sharp}B$ as its solution, we consider a cubic equation $X(A{\sharp}B)^{-1}X(A{\sharp}B)^{-1}X=C$ for A, B and C. The solution X = $(A{\sharp}B){\sharp}_{\frac{1}{3}}C$ is a candidate of the geometric mean of the three operators. However, this solution is not invariant under permutation unlike the geometric mean of two operators. To supply the lack of the property, we adopt a limiting process due to Ando-Li-Mathias. We define reasonable geometric means of k operators for all integers $k{\geq}2$ by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.

An Efficient and Fast Bit Allocation Algorithm for Multiuser OFDM Systems (다중 사용자 OFDM 시스템을 위한 효율적이고 빠른 비트 배정 알고리즘)

  • Kim, Min-Suk;Lee, Chang-Wook;Jeon, Gi-Joon
    • Proceedings of the KIEE Conference
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    • pp.218-220
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    • 2004
  • Orthogonal frequency division multiplexing(OFDM) is one of the most promising technique for next generation wireless broadband communication systems. In this paper, we propose a new bit allocation algorithm in multiuser OFDM. The proposed algorithm is derived from the geometric progression of the additional transmit power of subcarriers and the arithmetic-geometric means inequality. The simulation shows that this algorithm has similar performance to the conventional adaptive bit allocation algorithm and lower computational complexity than the existing algorithms.

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Sagae-Tanabe Weighted Means and Reverse Inequalities

  • Ahn, Eunkyung;Kim, Sejung;Lee, Hosoo;Lim, Yongdo
    • Kyungpook Mathematical Journal
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    • v.47 no.4
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    • pp.595-600
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    • 2007
  • In this paper we consider weighted arithmetic and geometric means of several positive definite operators proposed by Sagae and Tanabe and we establish a reverse inequality of the arithmetic and geometric means via Specht ratio and the Thompson metric on the convex cone of positive definite operators.

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