More Meaning From the Geometric Mean

2003 ◽  
Vol 96 (2) ◽  
pp. 142-146
Author(s):  
Bryan C. Dorner
Keyword(s):  
Human Activity ◽  
Mathematics Teacher ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
First Year ◽  
Geometric Meaning ◽  
Square Roots ◽  
Nature Of Mathematics ◽  
Basic Ideas ◽  
Universal Nature

We are all familiar with the average, or arithmetic mean, of two numbers. Less frequently used is the notion of the geometric mean. In “Geometric Meaning in the Geometric Mean Means More Meaningful Mathematics” in the March 2001 issue of the Mathematics Teacher, Matt E. Fluster shows how the geometric mean, s = _ab, of two positive numbers, a and b, can be used in a first-year algebra course to tie together geometric, algebraic, and computational investigations. In this article, I add a bit of history and an example suitable for more advanced courses. The example uses the geometric mean to compute square roots with Newton's method and does not require calculus. The history surrounding these concepts provides an opportunity to point out the universal nature of mathematics as a human activity. The basic ideas in this article were alive in the minds of ancient peoples who lived in what are now India, Pakistan, Iraq, and Egypt.

Exploring Geometric Maxima and Minima

1969 ◽  
Vol 62 (2) ◽  
pp. 85-90
Author(s):  
J. Garfunkel
Keyword(s):  
Mathematics Teacher ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Trigonometric Inequalities

In an Article that appeared in The Mathematics Teacher, * many interesting geometric as well as trigonometric inequalities were developed by starting with the theorem that the arithmetic mean of n positive quantities is not less than their geometric mean. The author also developed a few geometric maxima and minima properties in this manner.

Applications: Fast Brakes!

1989 ◽  
Vol 82 (2) ◽  
pp. 104-111
Author(s):  
David S. Daniels
Keyword(s):  
Mathematics Teacher ◽  
Traffic Accident ◽  
Police Department ◽  
Ninth Grade ◽  
First Year ◽  
Square Roots ◽  
Local Police ◽  
Good For ◽  
Ninth Grade Students ◽  
Algebra Class

As I was about to begin a new topic on square roots in my first-year algebra class, the question we have all heard popped up: “Is this stuff going to be good for anything?” Instead of some Pythagorean examples that are likely to be of more interest to a mathematics teacher than to ninth-grade students. I wanted a more compelling and forceful reply. Little did I know that help was on its way from the local police department and that I would soon discover enough about traffic-accident investigations to provide motivational applications of mathematics for my students.

On Statistics of Log-Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power

2012 ◽  
Vol E95-B (2) ◽  
pp. 647-650
Author(s):  
Ning WANG ◽  
Julian CHENG ◽  
Chintha TELLAMBURA
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Log Ratio

scholarly journals Faecal contamination of drinking water during collection and household storage: the need to extend protection to the point of use

2003 ◽  
Vol 1 (3) ◽  
pp. 109-115 ◽  
Author(s):  
Thomas F. Clasen ◽  
Andrew Bastable
Keyword(s):  
Drinking Water ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Faecal Contamination ◽  
Household Level ◽  
Safe Water ◽  
Female Head ◽  
Point Of Use ◽  
Thermotolerant Coliforms ◽  
And Storage

Paired water samples were collected and analysed for thermotolerant coliforms (TTC) from 20 sources (17 developed or rehabilitated by Oxfam and 3 others) and from the stored household water supplies of 100 households (5 from each source) in 13 towns and villages in the Kailahun District of Sierra Leone. In addition, the female head of the 85 households drawing water from Oxfam improved sources was interviewed and information recorded on demographics, hygiene instruction and practices, sanitation facilities and water collection and storage practices. At the non-improved sources, the arithmetic mean TTC load was 407/100 ml at the point of distribution, rising to a mean count of 882/100 ml at the household level. Water from the improved sources met WHO guidelines, with no faecal contamination. At the household level, however, even this safe water was subject to frequent and extensive faecal contamination; 92.9% of stored household samples contained some level of TTC, 76.5% contained more than the 10 TTC per 100 ml threshold set by the Sphere Project for emergency conditions. The arithmetic mean TTC count for all samples from the sampled households was 244 TTC per 100 ml (geometric mean was 77). These results are consistent with other studies that demonstrate substantial levels of faecal contamination of even safe water during collection, storage and access in the home. They point to the need to extend drinking water quality beyond the point of distribution to the point of consumption. The options for such extended protection, including improved collection and storage methods and household-based water treatment, are discussed.

Measles Immunity After Revaccination: Results in Children Vaccinated Before 10 Months of Age

PEDIATRICS ◽  
1982 ◽  
Vol 69 (3) ◽  
pp. 332-335
Author(s):  
Calvin C. Linnemann ◽  
Mark S. Dine ◽  
Gary A. Roselle ◽  
P. Anne Askey
Keyword(s):  
Measles Virus ◽  
Geometric Mean ◽  
Lymphocyte Transformation ◽  
First Year ◽  
Cell Mediated Immunity ◽  
Measles Vaccine ◽  
Pediatric Practice ◽  
Antibody Titers ◽  
Measles Vaccination ◽  
First Year Of Life

Measles immunity was studied in children in a private pediatric practice who had been revaccinated because they had received their primary measles vaccination before 1 year of age. Antibody was measured in 72 of these children who had received the first injection of live measles virus vaccine at <10 months of age, and the second at >1 year of age. Of the 72 children, 29 (40%) had no detectable antibody and the geometric mean titer for the group was approximately 1:4. Of the children with low antibody titers, 15 were given a third injection of measles vaccine and five (33%) still did not respond. Cell- mediated immunity as indicated by lymphocyte transformation to measles antigen was measured in 11 of the children. Five (45%) had responses to measles antigen, but the responses did not correlate with the presence or absence of antibody. This study confirms the observation that revaccination is unsuccessful in many children who received measles vaccine in the first year of life, and shows that even a third injection of vaccine may fail to produce a significant antibody response.

scholarly journals DUAL HESITANT FUZZY AGGREGATION OPERATORS

2015 ◽  
Vol 22 (2) ◽  
pp. 194-209 ◽  
Author(s):  
Dejian YU ◽  
Wenyu ZHANG ◽  
George HUANG
Keyword(s):  
Human Resources ◽  
Fuzzy Sets ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Aggregation Operators ◽  
Fuzzy Arithmetic ◽  
Hesitant Fuzzy Sets ◽  
Numerical Example ◽  
Fuzzy Environment ◽  
Fuzzy Aggregation

Dual hesitant fuzzy sets (DHFSs) is a generalization of fuzzy sets (FSs) and it is typical of membership and non-membership degrees described by some discrete numerical. In this article we chiefly concerned with introducing the aggregation operators for aggregating dual hesitant fuzzy elements (DHFEs), including the dual hesitant fuzzy arithmetic mean and geometric mean. We laid emphasis on discussion of properties of newly introduced operators, and give a numerical example to describe the function of them. Finally, we used the proposed operators to select human resources outsourcing suppliers in a dual hesitant fuzzy environment.

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.

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