• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.263-273
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• 2015

We derive modular equations of degree 2 to establish explicit relations for the parameterizations for the theta functions ${\varphi}$ and ${\psi}$. We then find specific values of the parameterizations to evaluate some new values of the modular j-invariant in terms of $J_n$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.529-531
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• 2011

We show that for all $N{\geq}1$, the modular function field $K(X_0^+(N))$ is generated by j(z)j(Nz) and j(z) + j(Nz) over ${\mathbb{C}}$, where j(z) is the modular invariant. Moreover we derive the defining equation of the the modular function field $K(X_0^+(N))$ from the classical modular polynomial ${\Phi}_N(X, Y )$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.919-920
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• 2011

For two even unimodular positive definite integral quadratic forms A[X], B[X] in n-variables, J. K. Koo [1, Theorem 1] showed that ${\theta}_A(\tau)/{\theta}_B(\tau)$ is a rational function of J, satisfying a certain condition. Where ${\theta}_A(\tau)$ and ${\theta}_B(\tau)$ are theta series related to A[X] and B[X], respectively, and J is the classical modular invariant. In this paper we give a simple proof of Theorem 1 of [1].

• Journal of the Chungcheong Mathematical Society
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• v.30 no.4
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• pp.445-447
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• 2017

We investigate congruence properties of Fourier coefficients of modular forms for ${\Gamma}^+_0(2)$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.379-383
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• 2007

We show that the modular equation ${\phi}^{T_n}_m$ (X, Y) for the Thompson series $T_n$ corresponding to ${\Gamma}_0$(n) gives an affine model of the modular curve $X_0$(mn) which has better properties than the one derived from the modular j invariant. Here, m and n are relative prime. As an application, we construct a ring class field over an imaginary quadratic field K by singular values of $T_n(z)\;and\;T_n$(mz).

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.603-613
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• 2006

Near-rings considered are right near-rings and R is a near-ring. $J_0^r(R)$, the right Jacobson radical of R of type-0, was introduced and studied by the present authors. In this paper $J_1^r(R)$ and $J_2^r(R)$, the right Jacobson radicals of R of type-1 and type-2 are introduced. It is proved that both $J_1^r$ and $J_2^r$ are radicals for near-rings and $J_0^r(R){\subseteq}J_1^r(R){\subseteq}J_2^r(R)$. Unlike the left Jacobson radical classes, the right Jacobson radical class of type-2 contains $M_0(G)$ for many of the finite groups G. Depending on the structure of G, $M_0(G)$ belongs to different right Jacobson radical classes of near-rings. Also unlike left Jacobson-type radicals, the constant part of R is contained in every right 1-modular (2-modular) right ideal of R. For any family of near-rings $R_i$, $i{\in}I$, $J_{\nu}^r({\oplus}_{i{\in}I}R_i)={\oplus}_{i{\in}I}J_{\nu}^r(R_i)$, ${\nu}{\in}\{1,2\}$. Moreover, under certain conditions, for an invariant subnear-ring S of a d.g. near-ring R it is shown that $J_2^r(S)=S{\cap}J_2^r(R)$.