• The Pure and Applied Mathematics
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• 제21권2호
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• pp.121-128
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• 2014

The Boolean rank of a nonzero m $m{\times}n$ 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. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with $1{\leq}k{\leq}min\{m,n\}$.

• Bulletin of the Korean Mathematical Society
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• 제48권5호
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• pp.969-990
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• 2011

Assume that X, partitioned into $2{\times}2$ block form, is a solution of the system of quaternion matrix equations $A_1XB_1$ = $C_1,A_2XB_2=C_2$. We in this paper give the maximal and minimal ranks of the submatrices in X, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse G, partitioned into $2{\times}2$ block form, of two quaternion matrices M and N. We present the formulas of the maximal and minimal ranks of the submatrices of G, and describe the properties of the submatrices of G as well. The findings of this paper generalize some known results in the literature.

• Bulletin of the Korean Mathematical Society
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• 제51권3호
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• pp.813-822
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• 2014

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

• Journal of the Korean Mathematical Society
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• 제42권4호
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• pp.871-881
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• 2005

In 1977, Ganter and Teirlinck proved that any $2t\;\times\;2t$ matrix with 2t nonzero elements can be partitioned into four sub-matrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any $mt{\times}nt$ matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. In 1995, Brualdi et al. showed that this conjecture is true if $m = 2,\;k\;\leq\;3\;or\;k\geq\;mn-2$. They also found a counterexample of this conjecture when m = 4, n = 4, k = 6 and t = 2. When t = 2, we show that this conjecture is true if $k{\leq}5$.

• Korean Journal of Cognitive Science
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• 제22권4호
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• pp.347-364
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• 2011

In this paper, we propose a automatic image bases extraction method for visual image reconstruction from brain activity using Non-negative Matrix Factorization (NMF). Image bases are basic elements to construct and present a visual image. Previous method used brain activity that evoked by predefined 361 image bases of four different sizes: $1{\times}1$, $2{\times}1$, $1{\times}2$, $2{\times}2$, and $2{\times}2$. Then the visual stimuli were reconstructed by linear combination of all the results from these image bases. While the previous method used 361 predefined image bases, the proposed method automatically extracts image bases which represent the image data efficiently. From the experiments, we found that the proposed method reconstructs the visual stimuli better than the previous method.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.485-495
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• 2011

Let $A_1$ and $A_2$ be $n{\times}n$ nonzero complex matrices, denote a linear combination of the two matrices by $A=c_1A_1+c_2A_2$, where $c_1$, $c_2$ are nonzero complex numbers. In this paper, we research the problem of the linear combinations in the general case. We give a sufficient and necessary condition for A is an involutive matrix and s+1-potent matrix, respectively, where $A_1$ is a tripotent matrix, with $A_1A_2=A_2A_1$. Then, using the results, we also give the sufficient and necessary conditions for the involutory of the linear combination A, where $A_1$ is a tripotent matrix, anti-idempotent matrix, and involutive matrix, respectively, and $A_2$ is a tripotent matrix, idempotent matrix, and involutive matrix, respectively, with $A_1A_2=A_2A_1$.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• 제47권1호
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• pp.75-80
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• 2019

In the recent aerospace and defence industries, it is required to develop small and low cost embedded systems. Based on a high speed digital signal processor (DSP), this paper first presents the development of an embedded system. To reduce the computation time of the high precision algorithm such as flight control, we also propose two algorithms for matrix multiplication. Validation results show that, compared to the performance using the $2{\times}2$ unit method, the performance of the proposed method 1 is improved, when the size of matrices is small. The proposed method 2 generally outperforms the $2{\times}2$ unit method.

• Bulletin of the Korean Mathematical Society
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• 제50권1호
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• pp.125-134
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• 2013

An element in a ring R is strongly clean provided that it is the sum of an idempotent and a unit that commutate. In this note, several necessary and sufficient conditions under which a $2{\times}2$ matrix over an integral domain is strongly clean are given. These show that strong cleanness over integral domains can be characterized by quadratic and Diophantine equations.

• Communications of the Korean Mathematical Society
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• 제29권2호
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• pp.227-237
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• 2014

For an $m{\times}n$ nonnegative integral matrix A, a generalized inverse of A is an $n{\times}m$ nonnegative integral matrix G satisfying AGA = A. In this paper, we characterize nonnegative integral matrices having generalized inverses using the structure of nonnegative integral idempotent matrices. We also define a space decomposition of a nonnegative integral matrix, and prove that a nonnegative integral matrix has a generalized inverse if and only if it has a space decomposition. Using this decomposition, we characterize nonnegative integral matrices having reflexive generalized inverses. And we obtain conditions to have various types of generalized inverses.

• Journal of the Korean Ceramic Society
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• 제30권11호
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• pp.897-904
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• 1993

CdS doped SiO2 glass coating films which are good candidates for the nonlinear optical materials were prepared by the Sol-Gel method. TEOS, C2H5OH, H2O and HCl were used as starting materials to obtain SiO2 matrix solutions. Then Cd(NO3)2.2H2O and CS(NH2)2 were dissolved into the SiO2 matrix solutions. Coating was performed several times in order to increase the thickness of coated film by the dip-coating method. Then heat treatments were carried out to control the size of CdS microcrystals doped in SiO2 glass matrix with respect to temperatures and times. CdS-doped SiO2 transparent coating films were successfully obtained. CdS crystals were changed from cubic to hexagonal type about $600^{\circ}C$.