• Journal of applied mathematics & informatics
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• 제35권1_2호
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• pp.191-204
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• 2017

A new fifth-order weighted Runge-Kutta algorithm based on heronian mean for solving initial value problem in ordinary differential equations is considered in this paper. Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge-Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge-Kutta(5,5) based on Contra Harmonic Mean and Runge-Kutta(5,5) based on Geometric Mean are carried out as well. The problems, methods and comparison criteria are specified very carefully. Numerical experiments show that the new algorithm performs better than other three methods in solving variety of initial value problems. The error analysis is discussed and stability polynomials and regions have also been presented.

• 대한수학회보
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• 제56권2호
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• pp.521-533
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• 2019

In this paper, we define a mean of two positive numbers called the k-golden mean and study some properties of it. Especially, we show that the 2-golden mean refines the harmonic and the geometric means. As an application, we define the k-golden ratio and give some properties of it as an generalization of the golden ratio. Furthermore, we define the matrix k-golden mean of two positive-definite matrices and give some properties of it. This is an improvement of Lim's results [2] for which the matrix golden mean.

• Kyungpook Mathematical Journal
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• 제54권4호
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• pp.521-529
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• 2014

In this paper we derive some optimal convex combination bounds related to arithmetic mean. We find the greatest values ${\alpha}_1$ and ${\alpha}_2$ and the least values ${\beta}_1$ and ${\beta}_2$ such that the double inequalities $${\alpha}_1T(a,b)+(1-{\alpha}_1)H(a,b)<A(a,b)<{\beta}_1T(a,b)+(1-{\beta}_1)H(a,b)$$ and $${\alpha}_2T(a,b)+(1-{\alpha}_2)G(a,b)<A(a,b)<{\beta}_2T(a,b)+(1-{\beta}_2)G(a,b)$$ holds for all a,b > 0 with $a{\neq}b$. Here T(a,b), H(a,b), A(a,b) and G(a,b) denote the second Seiffert, harmonic, arithmetic and geometric means of two positive numbers a and b, respectively.

• 대한수학회논문집
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• 제34권1호
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• pp.239-251
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• 2019

In this paper we introduce a new formulation of symmetric homogeneous bivariate means that depends on the variation of a given continuous strictly increasing function on (0, ${\infty}$). It turns out that this class of means includes a lot of known bivariate means among them the arithmetic mean, the harmonic mean, the geometric mean, the logarithmic mean as well as the first and second Seiffert means. Using this new formulation we introduce a lot of new bivariate means and derive some mean-inequalities.

• 대한수학교육학회지:학교수학
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• 제3권2호
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• pp.281-294
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• 2001

As a measure of the center of data, arithmetical mean, median, mode, harmonic mean and geometric mean are generally used. Students must learn qualitative aspect of average values as well as its calculation for its adequate use. As the result of the learning, they should be able to select the appropriate average value according to the characteristic of data and problem context. For this object, the historical origin and visual interpretation of average values were introduced in this paper. And to help teaching several meanings of average values, several examples including context were suggested.

• Kyungpook Mathematical Journal
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• 제49권4호
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• pp.595-603
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• 2009

We show that for an algebraic Riccati equation $X^*B^{-1}X-T^*X-X^*T=C$, its solutions are given by X = W + BT for some solution W of $X^*B^{-1}X$ = $C+T^*BT$. To generalize this, we give an equivalent condition for $\(\array{B&W\\W*&A}\)\;{\geq}\;0$ for given positive operators B and A, by which it can be regarded as Riccati inequality $X^*B^{-1}X{\leq}A$. As an application, the harmonic mean B ! C is explicitly written even if B and C are noninvertible.

• Communications for Statistical Applications and Methods
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• 제21권3호
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• pp.201-212
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• 2014

We provide a simple basic method to find bounds for higher order moments of unimodal distributions in terms of lower order moments when the random variable takes value in a given finite real interval. The bounds for moments in terms of the geometric mean of the distribution are also derived. Both continuous and discrete cases are considered. The bounds for the ratio and difference of moments are obtained. The special cases provide refinements of several well-known inequalities, such as Kantorovich inequality and Krasnosel'skii and Krein inequality.

• 대한수학회보
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• 제50권1호
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• pp.263-273
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• 2013

A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation $$\frac{f(x)-F(y)}{x-y}=M(g(x),\;G(y)),\;x{\neq}y$$, where M is a given mean and $f$, F, $g$, G are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.

• 한국의류학회지
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• 제15권4호
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• pp.431-435
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• 1991

The dispersion and Poiar force components of the surface free energy of PET fibers untreated and treated with hydrophilic chemicals, such as nonionic-soil release polymer (SRP), anionic, nonionic and hydrophilic silicone, were determined using harmonic-mean and geometric-mean methods. Contact angles of water and methylene iodide on the fibers were determined from the adhesion tensions using tensiometric method. Fibers treated with hydrophilic chemicals have the increased polar force component and the decreased dispersion force component. The adhesion tensions of triolein for the hydrophilic treated fibers were smaller than that for untreated fiber.

• Elastomers and Composites
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• 제39권1호
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• pp.12-22
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• 2004

라디오 주파수(13.56 MHz) 무전극 종형 플라즈마 반응기를 이용하여 천연고무 가교체의 표면을 클로로디플루오로메탄으로 처리하였다. FT-적외선 분광분석으로 표면개질 정도를 정성적으로 조사하였다. 플라즈마 처리표면의 마찰힘은 플라즈마 처리시간 증가에 따라 감소하였다. 고무표면에 에틸렌글리콜과 물을 떨어뜨려 접촉각을 측정한 결과 플라즈마 처리에 따라 감소하는 것으로 미루어 플라즈마 개질에 따라 표면극성이 증가하는 것을 확인하였다. 유리판 표면을 동일조건으로 플라즈마 처리한 경우는 극성의 감소만이 확인되었다. 표면자유에너지의 London 비극성 및 극성요소를 계산하는데 있어서 기하평균법과 조화평균법이 유용한 것으로 확인되었다. 평균방법에 관계없이 플라즈마 처리시간이 증가함에 따라 표면자유에너지는 증가하였다 그러나 조화평균법으로 계산된 자유에너지가 기하평균법으로 계산된 값에 비해 상대적으로 높았다. 플라즈마 표면개질은 마찰의 계면, 히스테리시스, 점성요소들에 영향을 미침으로써 마찰계수를 변화시키는 것으로 나타났다.