• Honam Mathematical Journal
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• v.43 no.3
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• pp.403-416
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• 2021

It is known that the rank 2 stably free syzygy module P⁽²⁾ is not free. This algebraic fact was proved analytically, but this remarkable fact still lacks of a simple algebraic proof. The main purpose of this paper is to give a partially algebraic proof by making use of a theorem whose proof is quite topological, and the further properties of the module will be discussed.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.685-689
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• 2005

We find some Hilbert functions of codimension 4, which are obtained from only k-configurations in $\mathbb{P}^{3}$ and support the $3^{rd}$ linear syzygy.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.335-343
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• 2021

In this article, we investigate the finiteness of Auslander Index when a ring A has not necessarily a canonical module, or a Gorenstein module. We also study the relations between column invariants and row invariants.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.705-717
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• 2018

We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.871-878
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• 2007

We define a numerical invariant $row_{CM}(A)$ over Cohen-Macaulay local ring A, which is related to rows of the presenting matrices of maximal Cohen-Macaulay modules without free summands. We show that $row(A)=row_{CM}(A)$ for a Cohen-Macaulay(not necessarily Gorenstein) local ring A.

• Journal of Integrative Natural Science
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• v.7 no.4
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• pp.278-282
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• 2014

We define a numerical invariant $row^*_j(A)$ over Cohen-Macaulay local ring A, which is related to the presenting matrices of the j-th syzygy module (with or without free summands). We show that $row_d(A)$=$row_{CM}(A)$ and $row^*_d(A)$=$row^*_{CM}(A)$ for a Cohen-Macaulay local ring A of dimension d.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1327-1338
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• 2015

In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a $Pr\ddot{u}fer$ v-multiplication domain if and only if w-w.gl.dim(R) ${\leq}1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).

• Journal for History of Mathematics
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• v.17 no.4
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• pp.37-44
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• 2004

Homological algebra has emerged and developed since 1950s. However, in 1890's Hilbert investigated the resolutions in his Syzygy Theorem which is a vital ingredient in homological algebra. In 1956 Serre has proved the finite global dimension of regular local rings. His result give a basic tool in homological algebra. This paper also deals with the depth formula that was raised by Auslander in 1961.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.1
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• pp.15-21
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• 2019

Let X be a reduced closed subscheme in ${\mathbb{P}}^n$ and $${\pi}_q:X{\rightarrow}Y={\pi}_q(X){\subset}{\mathbb{P}}^{n-1}$$ be an isomorphic projection from the center $q{\in}{\mathbb{P}}^n{\backslash}X$. Suppose that the minimal free presentation of $I_X$ is of the following form $$R(-3)^{{\beta}2,1}{\oplus}R(-4){\rightarrow}R(-2)^{{\beta}1,1}{\rightarrow}I_X{\rightarrow}0$$. In this paper, we prove that $H^1(I_X(k))=H^1(I_Y(k))$ for all $k{\geq}3$. This implies that Y is k-normal if and only if X is k-normal for $k{\geq}3$. Moreover, we also prove that reg(Y) ${\leq}$ max{reg(X), 4} and that $I_Y$ is generated by homogeneous polynomials of degree ${\leq}4$.

• Journal of Ecology and Environment
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• v.44 no.4
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• pp.207-211
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• 2020

Adult land crabs generally live on land while their larvae live in the sea. In the case of Sesarma haematoche, female crabs migrate from land to sea to release the larvae at the high tide of syzygy night. Artificial structures along coastal areas are being obstacles for the migration of land crabs and causing synchronized roadkills on coastal roads during breeding migration. In this research, we compared the sex ratios of crab populations in coastal areas with coastal roads and uninhabited island areas with no road. The proportion of females in inland habitats with coastal roads was significantly smaller than island habitats. In particular, females are exposed to the risk of annually repeated roadkills, and the proportion of females decreases rapidly with their growth. If this tendency is general for land crab populations in the coastal areas with roads, significant road mortality of female land crabs during breeding migration can lead to severe population decline in coastal areas. Therefore, it is necessary to take an action to save land crabs crossing coastal roads.