scholarly journals The matrix geometric mean of parameterized, weighted arithmetic and harmonic means

2011 ◽  
Vol 435 (9) ◽  
pp. 2114-2131 ◽  
Author(s):  
Sejong Kim ◽  
Jimmie Lawson ◽  
Yongdo Lim
Keyword(s):  
Geometric Mean ◽  
Weighted Arithmetic ◽  
The Matrix ◽  
Harmonic Means

On strengthened weighted Carleman's inequality

2003 ◽  
Vol 68 (3) ◽  
pp. 481-490 ◽  
Author(s):  
Aleksandra Čižmešija ◽  
Josip Pecarić ◽  
Lars–Erik Persson
Keyword(s):  
Geometric Mean ◽  
Weighted Arithmetic ◽  
Carleman’S Inequality ◽  
Carleman's Inequality

In this paper we prove a new refinement of the weighted arithmetic-geometric mean inequality and apply this result in obtaining a sharpened version of the weighted Carleman's inequality.

scholarly journals The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 380 ◽  
Author(s):  
Yongtao Li ◽  
Xian-Ming Gu ◽  
Jianxing Zhao
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Hölder Inequality ◽  
Power Mean ◽  
Weighted Arithmetic ◽  
Mathematical Equivalence ◽  
Power Mean Inequality ◽  
Mathematical Relations

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

Geospatial Suitability Indices Toolbox (GSI Toolbox)

2021 ◽  
Author(s):  
Christina Saltus ◽  
Todd Swannack ◽  
S. McKay
Keyword(s):  
Habitat Suitability ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Limiting Factor ◽  
Ecological Processes ◽  
Weighted Arithmetic ◽  
Suitability Index ◽  
Multiple Parameter ◽  
Spatially Distributed

Habitat suitability models are widely adopted in ecosystem management and restoration, where these index models are used to assess environmental impacts and benefits based on the quantity and quality of a given habitat. Many spatially distributed ecological processes require application of suitability models within a geographic information system (GIS). Here, we present a geospatial toolbox for assessing habitat suitability. The Geospatial Suitability Indices (GSI) toolbox was developed in ArcGIS Pro 2.7 using the Python® 3.7 programming language and is available for use on the local desktop in the Windows 10 environment. Two main tools comprise the GSI toolbox. First, the Suitability Index Calculator tool uses thematic or continuous geospatial raster layers to calculate parameter suitability indices based on user-specified habitat relationships. Second, the Overall Suitability Index Calculator combines multiple parameter suitability indices into one overarching index using one or more options, including: arithmetic mean, weighted arithmetic mean, geometric mean, and minimum limiting factor. The resultant output is a raster layer representing habitat suitability values from 0.0 to 1.0, where zero is unsuitable habitat and one is ideal suitability. This report documents the model purpose and development as well as provides a user’s guide for the GSI toolbox.

scholarly journals I. On a new method of approximation applicable to elliptic and ultra-elliptic functions

1860 ◽  
Vol 10 ◽  
pp. 473-475
Keyword(s):  
Geometric Mean ◽  
Elliptic Functions ◽  
New Method ◽  
Square Root ◽  
The Real ◽  
Rate Of Approximation ◽  
Real Application ◽  
Harmonic Means

I found my method on the known principle, that the geometric mean between two quantities is also a geometric mean between the arithmetic and harmonic means of those quantities. We may therefore approximate to the geometric mean of two quantities in this way:—Take their arithmetic and harmonic means; then take the arithmetic and harmonic means of those means; then of these last means again, and so on, as far as we please. If the ratio of the original quantities lies within the ratio of 1 : 2, the approximation proceeds with extraordinary rapidity, so that, in obtaining a fraction nearly equal to √2 by this method, we obtain a result true to eleven places of decimals at the fourth mean. I name this merely to show the rate of approximation. The real application of the method is to the integration of functions embracing a radical of the square root.

scholarly journals A Riemannian Limited-Memory BFGS Algorithm for Computing the Matrix Geometric Mean

2016 ◽  
Vol 80 ◽  
pp. 2147-2157 ◽  
Author(s):  
Xinru Yuan ◽  
Wen Huang ◽  
P.-A. Absil ◽  
K.A. Gallivan
Keyword(s):  
Geometric Mean ◽  
Limited Memory ◽  
The Matrix

scholarly journals On weighted means and their inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mustapha Raïssouli ◽  
Shigeru Furuichi
Keyword(s):  
Multilinear Algebra ◽  
Monotone Functions ◽  
Weighted Arithmetic ◽  
Related Operator ◽  
Weighted Means ◽  
Operator Version ◽  
Linear Multilinear Algebra ◽  
Operator Monotone Functions ◽  
Harmonic Means ◽  
Stabilizable Means

AbstractIn (Pal et al. in Linear Multilinear Algebra 64(12):2463–2473, 2016), Pal et al. introduced some weighted means and gave some related inequalities by using an approach for operator monotone functions. This paper discusses the construction of these weighted means in a simple and nice setting that immediately leads to the inequalities established there. The related operator version is here immediately deduced as well. According to our constructions of the means, we study all cases of the weighted means from three weighted arithmetic/geometric/harmonic means by the use of the concept such as stable and stabilizable means. Finally, the power symmetric means are studied and new weighted power means are given.

scholarly journals XIII. On a new method of approximation applicable to elliptic and ultra-elliptic functions

1860 ◽  
Vol 150 ◽  
pp. 223-227
Keyword(s):  
Geometric Mean ◽  
Elliptic Functions ◽  
New Method ◽  
Geometric Means ◽  
The Third ◽  
Harmonic Means ◽  
General Method

The difficulty of finding approximate values of elliptic functions of the third kind has led me to consider a general method of approximation, which I believe to be new, at least in its application to the evaluation of integrals of irrational functions. It depends on the known principle that the geometric mean between two quantities is also a geometric mean between their arithmetic and harmonic means. If we take any two positive quantities, we may approximate to their geometric means as follows:— Take the arithmetic and harmonic means of the two quantities, then again take the arithmetic and harmonic means of those means, and so on: the successive means will approximate with great rapidity to the geometric mean.

WEAK CONSISTENCY AND QUASI-LINEAR MEANS IMPLY THE ACTUAL RANKING

2002 ◽  
Vol 10 (03) ◽  
pp. 227-239 ◽  
Author(s):  
LUCIANO BASILE ◽  
LIVIA D'APUZZO
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Pairwise Comparisons ◽  
Relative Importance ◽  
Analytic Hierarchy ◽  
Left Eigenvector ◽  
The Matrix ◽  
Linear Means ◽  
The Right ◽  
Hierarchy Process

It is known that in the Analytic Hierarchy Process (A.H.P.) a scale of relative importance for alternatives is derived from a pairwise comparisons matrix A = (aij). Priority vectors are basically provided by the following methods: the right eigenvector method, the geometric mean method and the arithmetic mean method. Antipriority vectors can also be considered; they are built by both the left eigenvector method and mean procedures applied to the columns of A. When the matrix A is inconsistent, priority and antipriority vectors do not indicate necessarily the same ranking. We deal with the problem of the reliability of quantitative rankings and we use quasi-linear means for providing a more general approach to get priority and antipriority vectors.

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