• 대한수학회논문집
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• 제39권2호
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• pp.313-329
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• 2024

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ_≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.

• 대한수학회논문집
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• 제37권3호
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• pp.669-679
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• 2022

We provide a characterization of the positive monoids (i.e., additive submonoids of the nonnegative real numbers) that satisfy the finite factorization property. As a result, we establish that positive monoids with well-ordered generating sets satisfy the finite factorization property, while positive monoids with co-well-ordered generating sets satisfy this property if and only if they satisfy the bounded factorization property.

• Korean Journal of Mathematics
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• 제32권1호
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• pp.97-107
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• 2024

Let A ⊆ B be an extension of integral domains with characteristic zero. Let H(A, B) and h(A, B) be rings of composite Hurwitz series and composite Hurwitz polynomials, respectively. We simply call H(A, B) and h(A, B) composite Hurwitz rings of A and B. In this paper, we study when H(A, B) and h(A, B) are unique factorization domains (resp., GCD-domains, finite factorization domains, bounded factorization domains).

• 한국정보통신학회논문지
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• 제7권3호
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• pp.437-447
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• 2003

Circulant Matrix Factorization (CMF)는 covariance 행렬의 spectral factorization된 결과를 얻을 수 있다. 우리는 얻어진 결과를 가지고 일반적으로 잘 알려진 방법인 Schur algorithm을 이용하여 finite impulse response(FIR)와 infinite impulse response (IIR) lattice 필터를 설계하는 방법을 제안하였다. CMF는 기존에 많이 사용되는 root finding을 사용하지 않고 covariance polynomial로부터 minimum phase 특성을 가지는 polynomial을 얻는데 유용한 방법이다. 그리고 Schur algorithm은 toeplitz matrix를 빠르게 Cholesky factorization하기 위한 방법으로 이 방법을 이용하면 FIR/IIR lattice 필터의 계수를 쉽게 찾아낼 수 있다. 본 논문에서는 이러한 방법들을 이용하여 FIR과 IIR lattice 필터의 설계의 계산적인 예제를 제시했으며, 제안된 방법과 다른 기존에 제시되었던 방법 (polynomial root finding과 cepstral deconvolution)들과 성능을 비교 평가하였다.

• 대한수학회보
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• 제43권2호
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• pp.419-424
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• 2006

In this paper we extend the study of ascend and descend of factorization properties (for atomic domains, domains satisfying ACCP, bounded factorization domains, half-factorial domains, pre-Schreier and semirigid domains) to the finite factorization domains and idf-domains for domain extension $A\;{\subseteq}\;B$.

• 대한전자공학회논문지TC
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• 제41권1호
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• pp.35-44
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• 2004

Circulant Matrix Factorization (CMF)는 covariance 행렬의 spectral factorization된 결과를 얻을 수 있다. 우리는 얻어진 결과를 가지고 일반적으로 잘 알려진 방법인 Schur algorithm을 이용하여 finite impulse response (FIR)차 infinite impulse response (IIR) lattice 필터를 설계하는 방법을 제안하였다. CMF는 기존에 많이 사용되는 root finding을 사용하지 않고 covariance Polynomial로부터 minimum phase 특성을 가지는 polynomial을 얻는데 유용한 방법이다. 그리고 Schur algorithm은 toeplitz matrix를 빠르게 Cholesky factorization하기 위한 방법으로 이 방법을 이용하면 FIR/IIR lattice 필터의 계수를 쉽게 찾아낼 수 있다. 본 논문에서는 이러한 방법들을 이용하여 FIR과 IIR lattice 필터의 설계의 계산적인 예제를 제시했으며, 제안된 방법과 다른 기존에 제시되었던 방법 (polynomial root finding과 cepstral deconvolution)들과 성능을 비교 평가하였다.

• 정보보호학회논문지
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• 제9권3호
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• pp.3-12
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• 1999

공개키 암호법은 정수 인수분해의 어려움에 바탕을 둔 RSA와 이산대수문제의 어려움에 근거한 EIGamal 암호법을 대표된다. GF(qn)*에서 index-calculus 이산대수 알고리즘을 다항식 인수분해를 필요로 한다. 최근에 Niederreiter에 의하여 유한체위에서의 다항식 인수분해 알고리즘이 제안되었다. 이 논문에서는 정규기저(normal basis)를 이용한 유한체의 연산을 c-언어로 구현하고, 이것을 이용한 Niederreiter의 알고리즘을 기반으로 유한체위에서의 다항식 인수분해 알고리즘과 구현한 결과를 제시한다. The public key crytptosystem is represented by RSA based on the difficulty of integer factorization and ElGamal cryptosystem based on the intractability of the discrete logarithm problem in a cyclic group G. The index-calculus algorithm for discrete logarithms in GF(qn)* requires an polynomial factorization. The Niederreiter recently developed deterministic facorization algorithm for polynomial over GF(qn) In this paper we implemented the arithmetic of finite field with c-language and gibe an implementation of the Niederreiter's algorithm over GF(2n) using normal bases.

• 대한기계학회논문집A
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• 제27권2호
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• pp.276-284
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• 2003

The preconditioned conjugate gradient method is an efficient iterative solution scheme for large size finite element problems. As preconditioning method, we choose an incomplete Cholesky factorization which has efficiency and easiness in implementation in this paper. The incomplete Cholesky factorization mettled sometimes leads to breakdown of the computational procedure that means pivots in the matrix become minus during factorization. So, it is inevitable that a reduction process fur stabilizing and this process will guarantee robustness of the algorithm at the cost of a little computation. Recently incomplete factorization that enhances robustness through increasing diagonal dominancy instead of reduction process has been developed. This method has better efficiency for the problem that has rotational degree of freedom but is sensitive to parameters and the breakdown can be occurred occasionally. Therefore, this paper presents new method that guarantees robustness for this method. Numerical experiment shows that the present method guarantees robustness without further efficiency loss.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.19-35
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• 2004

We develop in this paper some preconditioners for sparse non-symmetric M-matrices, which combine a recursive two-level block I LU factorization with multigrid method, we compare these preconditioners on matrices arising from discretized convection-diffusion equations using up-wind finite difference schemes and multigrid orderings, some comparison theorems and experiment results are demonstrated.

• 정보보호학회논문지
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• 제1권1호
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• pp.97-101
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• 1991

다항식 인수분해 문제는 정수론에서 뿐만 아니라 Discrete logarithm과 관련하여 암호학의 응용에도 중요한 문제이다. Hensel의 Lifting Lemma를 이용하여 유한체위에서 다항식을 인수분해하여 정수계수위에서 다항식의 인수를 찾는 방법으로 정수계수위에서 다항식의 인수분해를 실행하였다.