• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.767-774
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• 2012

For a positive square-free integer $d$, let $t_d$ and $u_d$ be positive integers such that ${\epsilon}_d=\frac{t_d+u_d{\sqrt{d}}}{\sigma}$ is the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$, where ${\sigma}=2$ if $d{\equiv}1$ (mod 4) and ${\sigma}=1$ otherwise For a given positive integer $l$ and a palindromic sequence of positive integers $a_1$, ${\ldots}$, $a_{l-1}$, we define the set $S(l;a_1,{\ldots},a_{l-1})$ := {$d{\in}\mathbb{Z}|d$ > 0, $\sqrt{d}=[a_0,\overline{a_1,{\ldots},2a_0}]$}. We prove that $u_d$ < $d$ for all square-free integer $d{\in}S(l;a_1,{\ldots},a_{l-1})$ with one possible exception and apply it to Ankeny-Artin-Chowla conjecture and Mordell conjecture.

• Korean Journal of Mathematics
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• v.4 no.1
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• pp.7-16
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• 1996

The optimization problems with quadratic constraints often appear in various fields such as network flows and computer tomography. In this paper, we propose an algorithm for solving those problems and prove the convergence of the proposed algorithm.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1369-1374
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• 2014

We show that the Diophantine equation $$aQ(x_1,x_2)+bQ(x_3,x_4)+cQ(x_5,x_6)=abc$$ has integral solutions for arbitrary positive integers a, b, c when Q(x, y) is a norm form for some imaginary quadratic fields.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.431-452
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• 2022

For any positive integer 𝜇, we compute the mean value of the 𝜇-th derivative of quadratic Dirichlet L-functions over the rational function field 𝔽_q(t), where q is a power of 2.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1357-1365
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• 2013

We find a lower bound on the number of real/inert imagi-nary/ramified imaginary quadratic extensions of the function field $\mathbb{F}_q(t)$ whose ideal class groups have an element of a fixed order, where $q$ is a power of 2.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.28 no.4
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• pp.475-480
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• 2004

Function approximation is one of the most important and active research fields in design optimization. Accurate function approximations can reduce the repetitive computational effort fur system analysis. So this study presents an enhanced two-point diagonal quadratic approximation method. The proposed method is based on the Two-point Diagonal Quadratic Approximation method. But unlike TDQA, the suggested method has two quadratic terms, the diagonal term and the correction term. Therefore this method overcomes the disadvantage of TDQA when the derivatives of two design points are same signed values. And in the proposed method, both the approximate function and derivative values at two design points are equal to the exact counterparts whether the signs of derivatives at two design points are the same or not. Several numerical examples are presented to show the merits of the proposed method compared to the other forms used in the literature.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.83-89
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• 1998

Using the list of all imaginary quadratic fields with class number 1, 2, 3 and 6, we determine all imaginary bicyclic biquadratic number fields of class number 3. There are exactly 163 such fields and their conductors are less than or equal to 163 $\cdot$883.

• Bulletin of the Korean Mathematical Society
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• v.20 no.2
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• pp.81-85
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• 1983

It is the purpose of this paper to give some remarks on finite fields. We shall show that the little theorem of Fermat, Euler's criterion for quadratic residue mod p, and other few theorems in the number theory can be derived from the theorems in theory of finite field K=GF(p), where p is a prime. The forms of some irreducible ploynomials over K-GF(p) will be given explicitly.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.13-30
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• 2009

A subset of n tuples of elements of ${\mathbb{Z}}_9$ is said to be a code over ${\mathbb{Z}}_9$ if it is a ${\mathbb{Z}}_9$-module. In this paper we consider an special family of cyclic codes over ${\mathbb{Z}}_9$, namely quadratic residue codes. We define these codes in term of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over finite fields.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.907-928
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• 2015

We show in many cases that the Siegel-Ramachandra invariants generate the ray class fields over imaginary quadratic fields. As its application we revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of inequality argument together with Shimura's reciprocity law.