• East Asian mathematical journal
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• 제37권1호
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• pp.87-95
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• 2021

Let K be an imaginary quadratic field with ��K its ring of integers. For a positive integer N, let K(N) be the ray class field of K modulo N��K, and let ��N be the field of meromorphic modular functions of level N whose Fourier coefficients lie in the Nth cyclotomic field. For each h ∈ ��N, we construct a ray class invariant as its special value in terms of the extended form class group, and show that the invariant satisfies the natural transformation formula via the Artin map in the sense of Siegel and Stark. Finally, we establish an isomorphism between the extended form class group and Gal(K(N)/K) without any restriction on K.

• 대한수학회보
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• 제43권1호
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• pp.77-80
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• 2006

Let K be a number field, $K_n$ its ray class field with conductor n and L a Galois extension of K containing $K_n$. We prove that $L/K_n$ has a relative integral basis (RIB) under certain simple condition. Also we reduce the problem of the existence of a RIB to a quadratic extension of $K_n$ under some condition.

• 충청수학회지
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• 제34권4호
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• pp.317-326
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• 2021

Let K be an imaginary quadratic field other than ℚ(${\sqrt{-1}}$) and ℚ(${\sqrt{-3}}$), and let 𝒪_K be its ring of integers. Let N be a positive integer such that N = 5 or N ≥ 7. In this paper, we generate the ray class field modulo N𝒪_K over K by using a single x-coordinate of an elliptic curve with complex multiplication by 𝒪_K.

• East Asian mathematical journal
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• 제38권5호
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• pp.583-591
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• 2022

Let K be an imaginary quadratic field and 𝒪 be an order in K. Let H_𝒪 be the ring class field of 𝒪. Furthermore, for a positive integer N, let K_𝒪,N be the ray class field modulo N𝒪 of 𝒪. When the discriminant of 𝒪 is different from -3 and -4, we construct an extended form class group which is isomorphic to the Galois group Gal(K_𝒪,N/H_𝒪) and describe its Galois action on K_𝒪,N in a concrete way.

• 대한수학회지
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• 제50권4호
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• pp.847-864
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• 2013

By a change of variables we obtain new $y$-coordinates of elliptic curves. Utilizing these $y$-coordinates as meromorphic modular functions, together with the elliptic modular function, we generate the fields of meromorphic modular functions. Furthermore, by means of the special values of the $y$-coordinates, we construct the ray class fields over imaginary quadratic fields as well as normal bases of these ray class fields.

• Nuclear Engineering and Technology
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• 제54권12호
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• pp.4684-4692
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• 2022

Radionuclide identification is an important part of the nuclear material identification system. The development of artificial intelligence and machine learning has made nuclide identification rapid and automatic. However, many methods directly use existing deep learning models to analyze the gamma-ray spectrum, which lacks interpretability for researchers. This study proposes an explainable radionuclide identification algorithm based on the convolutional neural network and class activation mapping. This method shows the area of interest of the neural network on the gamma-ray spectrum by generating a class activation map. We analyzed the class activation map of the gamma-ray spectrum of different types, different gross counts, and different signal-to-noise ratios. The results show that the convolutional neural network attempted to learn the relationship between the input gamma-ray spectrum and the nuclide type, and could identify the nuclide based on the photoelectric peak and Compton edge. Furthermore, the results explain why the neural network could identify gamma-ray spectra with low counts and low signal-to-noise ratios. Thus, the findings improve researchers' confidence in the ability of neural networks to identify nuclides and promote the application of artificial intelligence methods in the field of nuclide identification.

• 대한수학회지
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• 제55권2호
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• pp.343-372
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• 2018

Let K be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$. Let E be an elliptic curve with complex multiplication by ${\mathcal{O}}_K$, and let $h_E$ be the Weber function on E. Let $N{\in}\{2,3,4,6\}$. We show that $h_E$ alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo $N{\mathcal{O}}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.

• 대한수학회지
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• 제52권5호
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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.

• East Asian mathematical journal
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• 제40권1호
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• pp.95-107
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• 2024

Let K be an imaginary quadratic field with ring of integers 𝓞_K and N be a positive integer. By K_(N) we mean the ray class field of K modulo N𝓞_K. In this paper, for each prime p of K relatively prime to N𝓞_K we explicitly describe the action of the Artin symbol (${\frac{K_{(N)}/K}{p}}$) on special values of modular functions of level N. Furthermore, we extend the Kronecker congruence relation for the elliptic modular function j to some modular functions of higher level.

• 대한수학회지
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• 제51권6호
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• pp.1269-1289
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• 2014

We first find a sufficient condition for a product of theta constants to be a Siegel modular function of a given even level. And, when $K_{(2p)}$ denotes the ray class field of $K=\mathbb{Q}(e^{2{\pi}i/5})$ modulo 2p for an odd prime p, we describe a subfield of $K_{(2p)}$ generated by the special value of a certain theta constant by using Shimura's reciprocity law.