• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.651-655
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• 1996

Lagrange's famous Four Square Theorem [L] says that every positive integer can be represented by the sum of four squares. This marvelous theorem was generalized by Mordell [M1] and Ko [K1] as follows : every positive definite integral quadratic form of two, three, four, and five variables is represented by the sum of five, six, seven, and eight squares, respectively. And they tried to extend this to positive definite integral quadratic forms of six or more variables.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.471-478
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• 2007

In this article we shall introduce several operators on the reproducing kernel spaces and investigate quadratic forms on the $\mathcal{l}^2$ space. Using these operators we shall obtain a particular solution of a system of quadratic equations(1.5). Finally we can find an approximate solution of(1.5) by optimization of a nonnegative biquadratic polynomial.

• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.577-586
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• 2006

In this paper we will propose the methods for finding the non-invertible ideals corresponding to non-primitive quadratic forms and clarify the structures of class SEMIGROUPS of imaginary quadratic orders which were given by Zanardo and Zannier [8], and we will give a general algorithm for calculating power of ideals/classes via the Dirichlet composition of quadratic forms which is applicable to cryptography in the class semigroup of imaginary quadratic non-maximal order and revisit the cryptosystem of Kim and Moon [5] using a Zanardo and Zannier [8]'s quantity as their secret key, in order to analyze Jacobson [7]'s revised cryptosystem based on the class semigroup which is an alternative of Kim and Moon [5]'s.

• Journal of the Korean Mathematical Society
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• v.56 no.3
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• pp.645-667
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• 2019

It has been little known when an Artinian point quotient has the strong Lefschetz property. In this paper, we find the Artinian point star configuration quotient having the strong Lefschetz property. We prove that if ${\mathbb{X}}$ is a star configuration in ${\mathbb{P}}^2$ of type s defined by forms (a-quadratic forms and (s - a)-linear forms) and ${\mathbb{Y}}$ is a star configuration in ${\mathbb{P}}^2$ of type t defined by forms (b-quadratic forms and (t - b)-linear forms) for $b=deg({\mathbb{X}})$ or $deg({\mathbb{X}})-1$, then the Artinian ring $R/(I{\mathbb_{X}}+I{\mathbb_{Y}})$ has the strong Lefschetz property. We also show that if ${\mathbb{X}}$ is a set of (n+ 1)-general points in ${\mathbb{P}}^n$, then the Artinian quotient A of a coordinate ring of ${\mathbb{X}}$ has the strong Lefschetz property.

• Journal of the Korean Mathematical Society
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• v.39 no.4
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• pp.525-542
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• 2002

In this article, we prove a certain local-global principle for representation of binary forms by an infinite family of quinary positive integral quadratic forms.

• East Asian mathematical journal
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• v.22 no.1
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• pp.71-77
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• 2006

By applying various known summation theorems to a general transformation formula based upon Bailey's transformation theorem due to Slater, Exton has obtained numerous and new quadratic transformations involving hypergeometric functions of order greater than two(some of which have typographical errors). We aim at first deriving a general quadratic transformation formula due to Exton and next providing a list of quadratic formulas(including the corrected forms of Exton's results) and some more results.

• Journal of the Korean Mathematical Society
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• v.38 no.3
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• pp.595-611
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• 2001

Let Q(n,1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[X] in Q(n,1), the theta series $\theta$(sub)A(z) = ∑(sub)X∈Z(sup)n e(sup)$\pi$izA[X] (Z∈h (※Equations, See Full-text) the complex upper half plane) is a modular form of weight n/2 for the congruence group Γ$_1$(8) = {$\delta$∈SL$_2$(Z)│$\delta$≡()mod 8} (※Equation, See Full-text). If n$\geq$24 and A[X], B{X} are tow quadratic forms in Q(n,1), the quotient $\theta$(sub)A(z)/$\theta$(sub)B(z) is a modular function for Γ$_1$(8). Since we identify the field of modular functions for Γ$_1$(8) with the function field K(X$_1$(8)) of the modular curve X$_1$(8) = Γ$_1$(8)＼h(sup)* (h(sup)* the extended plane of h) with genus 0, we can express it as a rational function of j(sub) 1,8 over C which is a field generator of K(X$_1$(8)) and defined by j(sub)1,8(z) = $\theta$$_3$(2z)/$\theta$$_3$(4z). Here, $\theta$$_3$ is the classical Jacobi theta series.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.67-80
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• 2019

For positive integers a, b, c, and an integer n, the number of integer solutions $(x,y,z){\in}{\mathbb{Z}}^3$ of $a{\frac{x(x-1)}{2}}+b{\frac{y(y-1)}{2}}+c{\frac{z(z-1)}{2}}=n$ is denoted by t(a, b, c; n). In this article, we prove some relations between t(a, b, c; n) and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in [6].

• Korean Journal of Mathematics
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• v.31 no.2
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• pp.181-188
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• 2023

In 1911, Dubouis determined all positive integers represented by sums of k nonvanishing squares for all k ≥ 4. As a generalization, Y.-S. Ji, M.-H. Kim and B.-K. Oh determined all positive definite binary quadratic forms represented by sums of k nonvanishing squares for all k ≥ 5. In this article, we give a simple proof for Ji-Kim-Oh's theorem for all k ≥ 10.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.219-225
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• 2008

In this paper, we obtain the general solution and the stability of the 2-dimensional vector variable quadratic functional equation f( x + y, z - w) + f(x - y, z + w) = 2f(x, z ) + 2f(y, ${\omega}$). The quadratic form f( x, y) = $ax^2$ + $by^2$ without cross product terms is a solution of the above functional equation.