• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1587-1598
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• 2018

We study additive decompositions (and generalized additive decompositions with a zero-dimensional scheme instead of a finite sum of rank 1 tensors), which are not of minimal degree (for sums of rank 1 tensors with more terms than the rank of the tensor, for a zero-dimensional scheme a degree higher than the cactus rank of the tensor). We prove their existence for all degrees higher than the rank of the tensor and, with strong assumptions, higher than the cactus rank of the tensor. Examples show that additional assumptions are needed to get the minimally spanning scheme of degree cactus +1.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1043-1056
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• 2008

The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix ordered pairs which satisfy multiplicative properties with respect to spanning column rank of matrices over semirings.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.301-312
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• 2008

The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix pairs which satisfy additive properties with respect to spanning column rank of matrices over semirings.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1841-1850
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• 2014

It is known that the ranks of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$ (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on $X_n={1,2,{\ldots},n}$, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of $\mathcal{SOP}_n$ and $\mathcal{SSPOP}_n$ are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$. Firstly, we characterize the structure of the minimal generating sets of $\mathcal{SOP}_n$. As applications, we obtain that the number of distinct minimal generating sets is $(n-1)^nn!$. Secondly, we show that, for $1{\leq}m{\leq}n-1$, the m-potent ranks of the semigroups $\mathcal{SOP}_n$ and $\mathcal{SPOP}_n$ are also n and 2n, respectively. Finally, we find that the 2-potent rank of $\mathcal{SSPOP}_n$ is n + 1.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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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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• v.42 no.3
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• pp.485-489
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• 2005

We show that any Drinfeld module of rank 2 on IF'q[T] over a global function field K has a global minimal Drinfeld module equation if and only if h(O$_{K}$) = 1.

• Korean Journal of Mathematics
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• v.10 no.1
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• pp.67-74
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• 2002

Suppose linear forms whose coefficients are arithmetic progressions are given. In this paper, we will find the upper and lower bounds of the minimal ranks of subforms of these forms which left the Frobenius numbers invariant. This is an improvement of Ritter's bounds.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.619-627
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• 2014

Let $\mathcal{M}(S)$ denote the set of all $m{\times}n$ matrices over a semiring S. For $A{\in}\mathcal{M}(S)$, zero-term rank of A is the minimal number of lines (rows or columns) needed to cover all zero entries in A. In [5], the authors obtained that a linear operator on $\mathcal{M}(S)$ preserves zero-term rank if and only if it preserves zero-term ranks 0 and 1. In this paper, we obtain new characterizations of linear operators on $\mathcal{M}(S)$ that preserve zero-term rank. Consequently we obtain that a linear operator on $\mathcal{M}(S)$ preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where $0{\leq}k{\leq}min\{m,n\}-1$ if and only if it strongly preserves zero-term rank h, where $1{\leq}h{\leq}min\{m,n\}$.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.521-525
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• 2005

We give a necessary and sufficient condition of a rationally elliptic space X such that the Dold-Lashof classifying space Baut1X for fibrations with the fiber X is rank one. It is only when X has the rational homotopy type of a sphere or the total space of a spherical fibration over a product of spheres.

• International Journal of Control, Automation, and Systems
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• v.3 no.1
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• pp.117-121
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• 2005

This paper deals with the state estimation of linear time-invariant discrete systems with unknown inputs. The forward sequences of the output are treated as additional outputs. In this case, the rank condition for designing the unknown input estimator is relaxed. The gain for minimal estimation error variance is presented, and a numerical example is given to verify the proposed unknown input estimator.