• 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.

• 대한수학교육학회지:학교수학
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• 제4권4호
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• pp.581-599
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• 2002

This study investigated the factorization concept development level of 3rd grades in middle school, the success of factorization problem solving, and the completion of factorization analogy tasks and science concepts analogy tasks. This study's results are followings. 1. Based on Sfard' reification levels, we classified students' factorization concept development levels from level 0 to level 3. As the students' development level was high, they tended to succeed the factorization problems gradually. 2. Experiencing factorization tasks which made students arrange factorization expressions hating same characterization, students ' factorization problem solving was improved. And, as the students' development level was high, they tended to attend to internal structural relations in factorization analogy tasks. 3. Analogy in factorization wasn't interrelated with analogy in science concepts. It said that analogy depended on the knowledges with it.

• 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.

• Journal of applied mathematics & informatics
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• 제26권3_4호
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• pp.779-784
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• 2008

We show that $\mathbb{Z}[\sqrt{-p}]$ is not a unique factorization domain (UFD) but a factorization domain (FD) with a condition $1\;+\;a^2p\;=\;qr$, where a and p are positive integers and q and r are positive primes in $\mathbb{Z}$ with q < p. Using this result, we also construct several specific non-unique factorization domains which are factorization domains. Furthermore, we prove that an integral domain $\mathbb{Z}[\sqrt{-p}]$ is not a UFD but a FD for some positive integer p.

• 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).

• 디지털콘텐츠학회 논문지
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• 제19권4호
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• pp.731-737
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• 2018

추천 시스템은 컨텐츠, 온라인 커머스, 소셜 네트워크, 광고 시스템 등 많은 분야에서 사용자가 관심 있을 만한 정보를 선별 제안함을 목적으로 활발하게 연구되고 있다. 그러나 과거 선호도 데이터를 기반으로 제안하는 추천시스템이 많고 과거 데이터가 적거나 없는 사용자를 대상으로는 제공하기 어려우므로 낮은 성능을 보인다는 부문에서 문제점이 있다. 따라서 더욱 고차원적인 데이터 분석에 관한 관심이 증가하고 있고 Matrix Factorization이 주목받고 있다. 이 논문은 그 중 추천시스템에서 주목받는 Factorization Machines Learning(FM)모델과 고차원 데이터 분석인 High-order Factorization Machines Learning(HOFM)의 비교와 재연을 연구하고 제안 한다.

• 경영과학
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• 제15권1호
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• pp.87-95
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• 1998

The computational speed of interior point method depends on the speed of Cholesky factorization. The supernodal column Cholesky factorization is a fast method that performs Cholesky factorization of sparse matrices with exploiting computer's characteristics. Three steps are necessary to perform the supernodal column Cholesky factorization : symbolic factorization, creation of the elimination tree, ordering by a post-order of the elimination tree and creation of supernodes. We study performing sequences of these three steps and efficient implementation of them.

• 대한수학회보
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• 제37권3호
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• pp.551-568
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• 2000

We propose new parallelizable block incomplete factorization preconditioners for a symmetric block-tridiagonal H-matrix. Theoretical properties of these block preconditioners are compared with those of block incomplete factorization preconditioners for the corresponding comparison matrix. Numerical results of the preconditioned CG(PCG) method using these block preconditioners are compared with those of PCG method using a standard incomplete factorization preconditioner to see the effectiveness of the block incomplete factorization preconditioners.

• 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.

• 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.