• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.577-581
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• 2002

The incidence matrices corresponding to a nil-algebra of finite index % can be used to determine the nilpotency. We find the smallest positive integer n such that the sum of the incidence matrices Σ$\_$p/$\^$p/ is invertible. In this paper, we give a different proof of the case that the nil-algebra of index 2 has nilpotency less than or equal to 4.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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• pp.311-318
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• 1999

Let A be an n$\times$n (0,1) matrix. Let f(A) denote the smallest nonnegative integer k such that per A[$\alpha$｜$\beta$]>0 and A($\alpha$｜$\beta$) is permutation equivalent to a lower triangular matrix for some $\alpha$, $\beta$$\in$Q\ulcorner,\ulcorner. In this case f(A) is called the feedback number of A. In this paper, feedback numbers of some maximal convertible (0,1) matrices are studied.

• Journal of Communications and Networks
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• v.3 no.4
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• pp.361-364
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• 2001

Over an additive abelian group G of order g and for a given positive integer $\lambda$, a generalized Hadamard matrix GH(g, $\lambda$) is defined as a gλ$\times$gλ matrix[h(i, j)], where 1 $\leq i \leqg\lambda and 1 \leqj \leqg\lambda$, such that every element of G appears exactly $\lambd$atimes in the list h($i_1, 1) -h(i_2, 1), h(i_1, 2)-h(i_2, 2), …, h(i_1, g\lambda) -h(i_2, g\lambda), for any i_1\neqi_2$. In this paper, we propose a new method of expanding a GH(g^m, \lambda_1) = B = [B_{ij}] over G^m$ by replacing each of its m-tuple B_{ij} with B_{ij} + GH(g, $\lambda_2) where m = g\lambda_2. We may use g^m/\lambda_1 (not necessarily all distinct) GH(g, \lambda_2$)s for the substitution and the resulting matrix is defined over the group of order g.

• Honam Mathematical Journal
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• v.44 no.4
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• pp.473-484
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• 2022

In this study, some new relations between generalized Fibonacci numbers and matrices are given. The work is designed in three stages: Firstly, it is obtained a relation between generalized Fibonacci numbers and integer powers of the matrices X satisfying the relation X² = pX +qI, and also, many results are derived from obtained relation. Then, it is established more general relation between generalized Fibonacci numbers and the square matrices X satisfying the condition X² = V_nX - (-q)ⁿI. Finally, some applications and numerical examples related to the obtained results are given.

• Industrial Engineering and Management Systems
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• v.16 no.1
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• pp.141-153
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• 2017

A binary integer programming model is proposed for a complex timetabling problem in a university faculty which conducts various degree programs. The decision variables are defined with fewer dimensions to economize the model size of large scale problems and to improve modeling efficiency. Binary matrices are used to incorporate the relationships between the courses and students, and the courses and teachers. The model includes generally applicable constraints such as completeness, uniqueness, and consecutiveness; and case specific constraints. The model was coded and solved using Open Solver which is an open-source optimizer available as an Excel add-in. The results indicate that complicated timetabling problems with large numbers of courses and student groups can be formulated more efficiently with fewer numbers of variables and constraints using the proposed modeling framework. The model could effectively generate timetables with a significantly lower number of work hours per week compared to currently used timetables. The model results indicate that the particular timetabling problem is bounded by the student overlaps, and both human and physical resource constraints are insignificant.

• JSTS:Journal of Semiconductor Technology and Science
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• v.7 no.4
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• pp.247-253
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• 2007

In this paper, a high throughput multiple transform architecture for H.264 Fidelity Range Extensions (FRExt) is proposed. New techniques are adopted which (1) regularize the $8{\times}8$ integer forward and inverse DCT transform matrices, (2) divide them into four $4{\times}4$ sub-matrices so that simple fast butterfly algorithm can be used, (3) because of the similarity of the sub-matrices, mixed butterflies are proposed that all the sub-matrices of $8{\times}8$ and matrices of $4{\times}4$ forward DCT (FDCT), inverse DCT (IDCT) and Hadamard transform can be merged together. Based on these techniques, a hardware architecture is realized which can achieve throughput of 1.488Gpixel/s when processing either $4{\times}4\;or\;8{\times}8$ transform. With such high throughput, the design can satisfy the critical requirement of the real-time multi-transform processing of High Definition (HD) applications such as High Definition DVD (HD-DVD) ($1920{\times}1080@60Hz$) in H.264/AVC FRExt. This work has been synthesized using Rohm 0.18um library. The design can work on a frequency of 93MHz and throughput of 1.488Gpixel/s with a cost of 56440 gates.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.625-636
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• 2015

The Boolean rank of a nonzero $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. We investigate the structure of linear transformations T : $\mathbb{M}_{m,n}{\rightarrow}\mathbb{M}_{p,q}$ which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, $2{\leq}k{\leq}$ min{m, n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.535-545
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• 2020

Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k for 1 ≤ k ≤ min{m, n} if and only if T is a (P, Q)-operator, that is, for fixed permutation matrices P and Q, T(A) = P AQ or, m = n and T(A) = P A^tQ for any m × n matrix A, where A^t is the transpose of A.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.439-448
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• 1994

It is known that the steepest descent(SD) method and the conjugate gradient(CG) method [1, 2, 5, 6] converge when these methods are applied to solve linear systems of the form Ax = b, where A is symmetric and positive definite. For some finite difference discretizations of elliptic problems, one gets positive definite matrices that are almost symmetric. Practically, the SD method and the CG method work for these matrices. However, the convergence of these methods is not guaranteed theoretically. The SD method is also called Orthores(1) in iterative method papers. Elman [4] states that the convergence proof for Orthores($\kappa$), with $\kappa$ a positive integer, is not heard. In this paper, we prove that the SD method and the CG method converge when the $\iota$$^2$ matrix norm of the non-symmetric part of a positive definite matrix is less than some value related to the smallest and the largest eigenvalues of the symmetric part of the given matrix.(omitted)

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.371-384
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• 2022

Fix an integer n ≥ 1. Then the simplex Π_n, Birkhoff polytope Ω_n, and Latin square polytope Λ_n each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ω_n are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.