• Journal of applied mathematics & informatics
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• v.35 no.1_2
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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.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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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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• v.54 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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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.

• School Mathematics
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• v.3 no.2
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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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• v.49 no.4
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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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• v.21 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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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.

• Journal of the Korean Society of Clothing and Textiles
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• v.15 no.4 s.40
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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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• v.39 no.1
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• pp.12-22
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• 2004

The plasma surface modification of natural rubber vulcanizate was carried out using chlorodifluoromethane in a radio-frequency (13.56 MHz) electrodeless bell type plasma reactor. The modification was qualitatively assessed by Fourier transform infrared spectroscopy. The frictional force of the plasma-treated surface was found to decrease with the time of plasma treatment. An increase in the surface polarity, as evidenced by the decrease in contact angle of a sessile drop of water and ethylene glycol on the natural rubber vulcanizate surface, was noted with the plasma modification. In the case of similar plasma treatment of glass surface, only a reduction in the polarity was observed. The use of geometric and harmonic mean methods was found to be useful to evaluate the London dispersive and specific components of surface free energy. Irrespective of the method used for evaluation, an increasing trend in the surface free energy was noted with increasing plasma treatment time. However, the harmonic mean method yielded comparatively higher values of surface free energy than the geometric mean method. The plasma surface modification was found to vary the frictional coefficient by influencing the interfacial, hysteresis and viscous components of friction in opposing dual manners.