• Korean Journal of Mathematics
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• v.32 no.1
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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).

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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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 $A\;{\subseteq}\;B$.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.7 no.3
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• pp.437-447
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• 2003

We propose the methods to design the finite impulse response (FIR) and the infinite impulse response (IIR) lattice filters using Schur algorithm through the spectral factorization of the covariance matrix by circulant matrix factorization (CMF). Circulant matrix factorization is also very powerful tool used fur spectral factorization of the covariance polynomial in matrix domain to obtain the minimum phase polynomial without the polynomial root finding problem. Schur algorithm is the method for a fast Cholesky factorization of Toeplitz matrix, which easily determines the lattice filter parameters. Examples for the case of the FIR Inter and for the case of the IIR filter are included, and performance of our method check by comparing of our method and another methods (polynomial root finding and cepstral deconvolution).

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.407-464
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• 2023

Let R be a commutative ring with identity. The structure theorem says that R is a PIR (resp., UFR, general ZPI-ring, π-ring) if and only if R is a finite direct product of PIDs (resp., UFDs, Dedekind domains, π-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations v or t as follows: An integral domain R is a Krull domain if and only if every nonzero proper principal ideal of R can be written as a finite v- or t-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation u on R, so that R is a general Krull ring if and only if every proper principal ideal of R can be written as a finite u-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

• Journal of the Institute of Electronics Engineers of Korea TC
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• v.41 no.1
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• pp.35-44
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• 2004

We Propose the methods to design the finite impulse response (FIR) and the infinite impulse response (IIR) lattice filters using Schur algorithm through the spectral factorization of the covariance matrix by circulant matrix factorization (CMF). Circulant matrix factorization is also very powerful tool used for spectral factorization of the covariance polynomial in matrix domain to obtain the minimum phase polynomial without the polynomial root finding problem. Schur algorithm is the method for a fast Cholesky factorization of Toeplitz matrix, which easily determines the lattice filter parameters. Examples for the case of the FIR filter and for the case of the In filter are included, and performance of our method check by comparing of our method and another methods (polynomial root finding and cepstral deconvolution).

• Journal of the Computational Structural Engineering Institute of Korea
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• v.32 no.2
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• pp.117-124
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• 2019

Parallel sparse solvers are essential for solving large-scale finite element models. This paper introduces the combination of iterative and direct solver that can be applied efficiently to problems that require continuous solution for a subtly changing sequence of systems of equations. The iterative-direct sparse solver combination technique, proposed and implemented in the parallel sparse solver package, PARDISO, means that iterative sparse solver is applied for the newly updated linear system, but it uses the direct sparse solver's factorization of previous system matrix as a preconditioner. If the solution does not converge until the preset iterations, the solution will be sought by the direct sparse solver, and the last factorization results will be used as a preconditioner for subsequent updated system of equations. In this study, an improved method that sets the maximum number of iterations dynamically at the first Krylov iteration step is proposed and verified thereby enhancing calculation efficiency by the frequency domain analysis.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.4 no.2
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• pp.108-120
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• 1992

An ADI model for linearized shallow water equation is modified using the method of factorization. In order to show its validity. the computational results are compared both with the analytical solution and with those from existing models, for a rectangualr domain with constant and varying amplitudes at the open boundary. It is shown the accuracy of numerical solutions depends on the size of time step. depth and bottom friction. The modified ADI model is shown to be superior to the existing models such as Leendertse (1971). Butler (1980) and Sheng (1983).

• Bulletin of the Korean Mathematical Society
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• v.20 no.1
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• pp.45-49
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• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.