• Korean Journal of Mathematics
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• v.9 no.1
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• pp.67-73
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• 2001

For some equation related with exponential function, we seek roots and find the properties of the roots. By using the relation of the roots and attractors, we find a region in the basin of attraction of the attractor at infinity for Newton's method for solving given equation.

• Journal of Microbiology and Biotechnology
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• v.13 no.5
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• pp.641-646
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• 2003

An accurate and efficient method for measuring the mass of hairy roots using conductometry is established. A conductivity equation expressed in terms of the concentration of the ion species in the medium is suggested. By using this equation, the effect of the individual ions on the total conductivity can be quantitatively analyzed. An equation for the in situ estimation of the cell growth coefficient for determining the mass of hairy roots is established based on measurements of the nitrogen concentration and conductivity during cultivation. The proposed equation does not require preliminary experiments to determine the cell growth coefficient. Instead, the physiological characteristics of the plant species are reflected by introducing the cellular nitrogen content. Since the cell growth coefficient is determined by measuring the major ionic nutrient concentrations, it is more effective to express the dynamics of an actual culture system. This improved method for determining the mass of hairy roots was successfully utilized in a fed-batch culture system.

• Journal for History of Mathematics
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• v.27 no.3
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• pp.165-182
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• 2014

After algebraic expressions for the roots of 3rd and 4th degree polynomial equations were given in the mid 16th century, seeking such a formula for the 5th and greater degree equations had been one main problem for algebraists for almost 200 years. Lagrange made careful and thorough investigation of various solving methods for equations with the purpose of finding a principle which could be applicable to general equations. In the process of doing this, he found a relation between the roots of the original equation and its auxiliary equation using permutations of the roots. Lagrange's ingenious idea of using permutations of roots of the original equation is regarded as the key factor of the Abel's proof of unsolvability by radicals of general 5th degree equations and of Galois' theory as well. This paper intends to examine Lagrange's contribution in the theory of polynomial equations, providing a detailed analysis of various solving methods of Lagrange and others before him.

• Journal of the Korean Statistical Society
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• v.23 no.1
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• pp.13-32
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• 1994

Let A(n) and B(n) be sequences of $m \times m$ random matrices with a joint asymptotic distribution as $n \to \infty$. The asymptotic distribution of the ordered roots of $$\mid$A(n) - f B(n)$\mid$ = 0$ depends on the multiplicity of the roots of a determinatal equation involving parameter roots. This paper treats the asymptotic distribution of the roots of the above determinantal equation in the case where some of parameter roots are zero. Furthermore, we apply our results to deriving the asymptotic distributions of the eigenvalues of the MANOVA matrix in the noncentral case when the underlying distribution is not multivariate normal and some parameter roots are zero.

• Current Optics and Photonics
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• v.4 no.4
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• pp.326-329
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• 2020

The Eaton lens, with spherical symmetry to its refractive index, was described by Eaton in 1952 and was found recently in the design of an invisible sphere for cloaking. In this paper, an Eaton lens for rotating 90° was designed using Luneburg theory, by which we found it was a fourth-order equation in the refractive index n. Therefore, the refractive index n has four roots. The equation in n was solved and studied using mathematical technology. The unsuitable complex roots of the equation should be dropped; consequently, only one of the four roots remained. To verify the refractive-index profile, the only root was solved for, before a simulation using finite-element analysis (FEA) was performed. The simulation showed that all rays will bend 90° to the right. The result of the simulation is identical to our expectation. This treatment provides a possible method for rotating light at many other angles.

• The Pure and Applied Mathematics
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• v.17 no.2
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• pp.185-188
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• 2010

In this note the distribution of roots of cubic equations in contrast to 0 is given, which is useful to discuss eigenvalues for qualitative properties of differential equations.

• Communications of Mathematical Education
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• v.29 no.4
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• pp.699-722
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• 2015

Based on Lagrange's general theory of algebraic equations, by applying the solution of the equation using the relationship between roots and coefficients to the high school 1st grade class, the purpose of this study is to recognize the significance of symmetry associated with the solution of the equation. Symmetry is the core idea of Lagrange's general theory of algebraic equations, and the relationship between roots and coefficients is an important means in the solution. Through the lesson, students recognized the significance of learning about the relationship between roots and coefficients, and understanded the idea of symmetry and were interested in new solutions. These studies gives not only the local experience of solutions of the equations dealing in school mathematics, but the systematics experience of general theory of algebraic equations by the didactical organization, and should be understood the connections between knowledges related to the solutions of the equation in a viewpoint of the mathematical history.

• Communications of Mathematical Education
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• v.22 no.4
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• pp.505-514
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• 2008

In this paper we study on derivation of formulas for roots of quadratic equation, cubic equation, and quartic equation through method analogy. Our argument is based on the norm form of polynomial. We also present some mathematical content knowledge related with main discussion of this article.

• Journal of Korean Society of Forest Science
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• v.96 no.2
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• pp.138-144
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• 2007

This study was undertaken to compare the biomass of Abies koreana growing at two sites. A $10{\times}10m$ plot was established in each site of a natural stand in Mt. Jiri and a plantation in Gyeongsan nursery. Five trees of A. koreana were randomly selected in each site. The following traits were investigated from each tree : height, basal diameter, age, weight of stem, branches, and needles as above-ground traits and weight of total roots, horizontal roots, and vertical roots as below-ground traits. In Gyeongsan nursery, age of sample trees was negatively correlated with both height and weight of total stem, while height was highly correlated with weight of horizontal roots. There was high correlation between the basal diameter and weight of total stem, and between the basal diameter and weight of roots. In Mt. Jiri stand, most of the above-ground traits except age were significantly correlated with the below-ground traits. The linear regression equation between the cross section area of base (X) and the weight of total stem (Y) in Gyeongsan nursery was Y=12.66X-12.92, and correlation was significant ($R^2=0.89$). The linear regression equation between the cross section area of base(X) and the weight of total branches (Y) in Mt. Jiri stand was Y=25.51X+6.00, and correlation was highly significant ($R^2=1.0$).

• Journal for History of Mathematics
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• v.25 no.1
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• pp.15-27
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• 2012

Thomas Harriot(1560-1621) introduced a simplified notation for algebra. His fundamental research on the theory of equations was far ahead of that time. He invented certain symbols which are used today. Harriot treated all answers to solve equations equally whether positive or negative, real or imaginary. He did outstanding work on the solution of equations, recognizing negative roots and complex roots in a way that makes his solutions look like a present day solution. Since he published no mathematical work in his lifetime, his achievements were not recognized in mathematical history and mathematics education. In this paper, by comparing his works with Viete and Descartes those are mathematicians in the same age, I show his achievements in mathematics.