A serendipitous path to a famous inequality

2008 ◽  
Vol 92 (523) ◽  
pp. 50-54
Author(s):  
Robert M. Young
Keyword(s):  
Real Number ◽  
Geometric Means ◽  
Algebraic Identity ◽  
Binomial Expansion

The mysterious path of discovery – the tireless experimentation in search of patterns, the veiled connections that suddenly unfold, serendipity – all these elements combine to make mathematics so magical. The purpose of this note is to show how a routine algebraic identity, the binomial expansion of (x- 1)2, can be used to give a new proof of the fundamental inequality between the arithmetic and geometric means. The proof will provide further evidence that a great deal of useful mathematics can be derived from the obvious assertion that the square of a real number is never negative.

Real number arithmetic for mixed behavioural and structural descriptions

1990 ◽  
Vol 137 (6) ◽  
pp. 446
Author(s):  
M.G. Hill ◽  
N.E. Peeling ◽  
I.F. Currie ◽  
J.D. Morison ◽  
E.V. Whiting ◽  
...  
Keyword(s):  
Real Number ◽  
Structural Descriptions

Estimation of geographical variation of cancer risks in drinking water in Japan

2006 ◽  
Vol 6 (2) ◽  
pp. 31-37
Author(s):  
K. Ohno ◽  
E. Kadota ◽  
Y. Kondo ◽  
T. Kamei ◽  
Y. Magara
Keyword(s):  
Geographical Variation ◽  
Statistical Data ◽  
Raw Water ◽  
Cancer Risks ◽  
Geometric Means ◽  
Toxicological Effects ◽  
Total Cancer ◽  
Big Cities ◽  
The Relationship ◽  
Plausible Reason

The cancer risks posed by ten substances in raw and purified water were estimated for each municipality in Japan to compare risks between raw and purified water, and inter-municipality. Water concentrations were estimated by use of statistical data. Assigning cancer unit risks to each substance and applying the assumption of additive toxicological effects to multiple carcinogens, total cancer risks of the waters were estimated. As a result, the geometric means of total cancer risks in raw and purified water were 1.16×10−5 and 2.18×10−5, respectively. In raw water, the contribution ratio of arsenic to total cancer risk accounted for 97%. In purified water, that of four trihalomethanes (THMs) accounted for 54%. The increase of total cancer risks in purified water was due to THMs. In regard to the geographical variation, the relationship between population size and total cancer risks were investigated. The result was that there were higher cancer risks in the big cities with the population more than a million both in raw and purified water. One plausible reason for the higher risks in purified water in the big cities is a larger chlorination dose due to the huge water supply areas. The reason for the increase in raw water remained unclear.

scholarly journals Mathematical Properties of Variable Topological Indices

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 43
Author(s):  
José M. Sigarreta
Keyword(s):  
Real Number ◽  
Large Class ◽  
Symmetric Functions ◽  
Topological Indices ◽  
Current Interest ◽  
Zagreb Indices ◽  
Mathematical Properties ◽  
Connectivity Indices ◽  
Optimal Bounds ◽  
New Framework

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

scholarly journals Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 32
Author(s):  
Pragati Gautam ◽  
Luis Manuel Sánchez Ruiz ◽  
Swapnil Verma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Real Number ◽  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Rectangular Metric Space ◽  
New Type ◽  
Symmetry Requirement ◽  
Definition Of

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result.

scholarly journals A Local Barycentric Version of the Bak–Sneppen Model

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Philip Kennerberg ◽  
Stanislav Volkov
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

AbstractWe study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let $$N\ge 3$$ N ≥ 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $$\zeta $$ ζ . We show that in case where $$\zeta $$ ζ is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

Microbiological Quality of Fresh Blue Crabmeat, Clams and Oysters

1983 ◽  
Vol 46 (11) ◽  
pp. 978-981 ◽  
Author(s):  
B. A. WENTZ ◽  
A. P. DURAN ◽  
A. SWARTZENTRUBER ◽  
A. H. SCHWAB ◽  
R. B. READ
Keyword(s):  
Escherichia Coli ◽  
Staphylococcus Aureus ◽  
Microbiological Quality ◽  
Fecal Coliforms ◽  
Geometric Means ◽  
Colony Forming Units ◽  
Plate Counts ◽  
The Mean ◽  
Eastern Oysters

The microbiological quality of fresh blue crabmeat, soft- and hardshell clams and shucked Eastern oysters was determined at the retail (crabmeat, oysters) and wholesale (clams) levels. Geometric means of aerobic plate counts incubated at 35°C were: blue crabmeat 140,000 colony-forming units (CFU)/g, hardshell clams, 950 CFU/g, softshell clams 680 CFU/g and shucked Eastern oysters 390,000 CFU/g. Coliform geometric means ranged from 3,6/100 g for hardshell clams to 21/g for blue crabmeat. Means for fecal coliforms or Escherichia coli ranged from <3/100 g for clams to 27/100 g for oysters, The mean Staphylococcus aureus count in blue crabmeat was 10/g.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

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