Remarks on the Neglected Mean

1973 ◽  
Vol 66 (3) ◽  
pp. 253-255
Author(s):  
Joseph L. Ercolano
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Real Numbers ◽  
Positive Real ◽  
Golden Proportion ◽  
The One

The harmonic mean of two positive, real numbers was known to early Greek mathematicians. In fact, it is alleged that “Pythagoras learned in Mesopotamia of three means—the arithmetic, the geometric, and the subcontrary (later called the harmonic)—and of the ‘golden proportion’ relating two of these: the first of two numbers is to their arithmetic mean as their harmonic mean is to the second of the numbers” (Boyer 1968). Archytas, a disciple of Pythagoras (whose most important contribution to mathematics may very well have been his intervention with Dionysius to save the life of his friend, Plato), wrote on the application of these three means to music, and is possibly the one who is responsible for renaming the suhcontrary mean the harmonic mean (Boyer 1968).

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

2020 ◽  
Vol 70 (5) ◽  
pp. 1097-1112
Author(s):  
Hong-Hu Chu ◽  
Tie-Hong Zhao ◽  
Yu-Ming Chu
Keyword(s):  
Elliptic Integral ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Complete Elliptic Integral ◽  
Quadratic Mean ◽  
Sharp Bounds ◽  
Toader Mean ◽  
The One

AbstractIn the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities$$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

Intuitionistic fuzzy two-factor variance analysis of movie ticket sales

2021 ◽  
pp. 1-11
Author(s):  
Velichka Traneva ◽  
Stoyan Tranev
Keyword(s):  
Comparative Analysis ◽  
Real Data ◽  
Intuitionistic Fuzzy Sets ◽  
Real Numbers ◽  
Data Set ◽  
Important Method ◽  
Ticket Sales ◽  
Intuitionistic Fuzzy ◽  
The One ◽  
First Time

Analysis of variance (ANOVA) is an important method in data analysis, which was developed by Fisher. There are situations when there is impreciseness in data In order to analyze such data, the aim of this paper is to introduce for the first time an intuitionistic fuzzy two-factor ANOVA (2-D IFANOVA) without replication as an extension of the classical ANOVA and the one-way IFANOVA for a case where the data are intuitionistic fuzzy rather than real numbers. The proposed approach employs the apparatus of intuitionistic fuzzy sets (IFSs) and index matrices (IMs). The paper also analyzes a unique set of data on daily ticket sales for a year in a multiplex of Cinema City Bulgaria, part of Cineworld PLC Group, applying the two-factor ANOVA and the proposed 2-D IFANOVA to study the influence of “ season ” and “ ticket price ” factors. A comparative analysis of the results, obtained after the application of ANOVA and 2-D IFANOVA over the real data set, is also presented.

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.

Determinacy of Banach games

1985 ◽  
Vol 50 (1) ◽  
pp. 110-122
Author(s):  
Howard Becker
Keyword(s):  
Winning Strategy ◽  
Similar Type ◽  
Infinite Games ◽  
Real Numbers ◽  
Positive Real ◽  
Infinite Game ◽  
First Time

For any A ⊂ R, the Banach game B(A) is the following infinite game on reals: Players I and II alternately play positive real numbers a1; a2, a3, a4,… such that for n > 1, an < an−1. Player I wins iff ai exists and is in A.This type of game was introduced by Banach in 1935 in the Scottish Book [15], Problem 43. The (rather vague) problem which Banach posed was to characterize those sets A for which I (II) has a winning strategy in B(A). (There are three parts to Problem 43. In the first, Mazur defined a game G**(A) for every set A ⊂ R and conjectured that II has a winning strategy in G**(A) iff A is meager and I has a winning strategy in G**(A) iff A is comeager in some neighborhood; this conjecture was proved by Banach. Presumably Banach had this result in mind when he asked the question about B(A), and hoped for a similar type of characterization.) Incidentally, Problem 43 of the Scottish Book appears to be the first time infinite games of any sort were studied by mathematicians.This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none. This paper is concerned with a different (also rather vague) question: For which sets A is the Banach game B(A) determined? To say that B(A) is determined means, of course, that one of the players has a winning strategy for B(A).

scholarly journals On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Dağistan Simsek ◽  
Bilal Demir ◽  
Cengiz Cinar
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Positive Real

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.

A Neutral Model With Fluctuating Population Size and Its Effective Size

Genetics ◽  
2002 ◽  
Vol 161 (1) ◽  
pp. 381-388
Author(s):  
Masaru Iizuka ◽  
Hidenori Tachida ◽  
Hirotsugu Matsuda
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Diffusion Model ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Effective Size ◽  
Neutral Model ◽  
Average Heterozygosity ◽  
Special Case ◽  
Concrete Model

Abstract We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

2014 ◽  
Vol 33 (2) ◽  
pp. 59-67
Author(s):  
Pankaj Kumar ◽  
S. S. Bhatia ◽  
Vijay Kumar
Keyword(s):  
Statistical Convergence ◽  
Convergence Criteria ◽  
Double Sequences ◽  
Normed Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Probabilistic Normed Spaces

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$  such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

A FUNCTIONAL INEQUALITY FOR THE GAMMA FUNCTION

2013 ◽  
Vol 11 (02) ◽  
pp. 1350010
Author(s):  
HORST ALZER
Keyword(s):  
Gamma Function ◽  
Functional Inequality ◽  
Real Numbers ◽  
Positive Real ◽  
Euler’S Constant ◽  
Euler's Constant

Let α and β be real numbers. We prove that the functional inequality [Formula: see text] holds for all positive real numbers x and y if and only if [Formula: see text] Here, γ denotes Euler's constant.

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