• 대한수학회보
• /
• 제52권1호
• /
• pp.263-273
• /
• 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 $J_n$.

• 충청수학회지
• /
• 제24권3호
• /
• pp.529-531
• /
• 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 ${\Phi}_N(X, Y )$.

• 충청수학회지
• /
• 제30권4호
• /
• pp.445-447
• /
• 2017

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

• 충청수학회지
• /
• 제24권4호
• /
• pp.919-920
• /
• 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].

• 대한수학회지
• /
• 제47권2호
• /
• pp.299-309
• /
• 2010

The Weyl group of a compact connected Lie group is a reflection group. If such Lie groups are locally isomorphic, the representations of the Weyl groups are rationally equivalent. They need not however be equivalent as integral representations. Turning to the invariant theory, the rational cohomology of a classifying space is a ring of invariants, which is a polynomial ring. In the modular case, we will ask if rings of invariants are polynomial algebras, and if each of them can be realized as the mod p cohomology of a space, particularly for dihedral groups.

• 충청수학회지
• /
• 제26권1호
• /
• pp.213-219
• /
• 2013

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, we compute the Galois actions of a class invariant from a generalized Weber function $g_1$ over imaginary quadratic number fields with discriminant $D{\equiv}64(mod72)$.

• 충청수학회지
• /
• 제24권4호
• /
• pp.921-925
• /
• 2011

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of a class invariant from a generalized Weber function $g_2$ over quadratic number fields with discriminant $D{\equiv}21$ (mod 36).

• 충청수학회지
• /
• 제23권4호
• /
• pp.853-860
• /
• 2010

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of a class invariant from a generalized Weber function $g_2$ over quadratic number fields with discriminant $D{\equiv}-3$ (mod 36).

• 대한수학회보
• /
• 제44권2호
• /
• pp.379-383
• /
• 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).

• 전자공학회논문지C
• /
• 제35C권2호
• /
• pp.71-79
• /
• 1998

In this paper, we propose a new mART(modified ART) neural network by combining the winner neuron definition method of SOM(self-organizing map) and the real-time adaptive clustering function of ART(adaptive resonance theory) and construct it in a modular structure, for the purpose of organizing the feature maps of three dimensional targets. Being constructed in a modular structure, the proposed modular mART can effectively prevent the clusters from representing multiple classes and can be trained to organze two dimensional distortion invariant feature maps so as to recognize targets with three dimensional distortion. We also present the recognition result and self-organization perfdormance of the proposed modular mART neural network after carried out some experiments with 14 tank and fighter target models.