• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.545-550
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• 2005

We find a necessary and sufficient condition that the number of some Betti numbers and shifts appear in the last free module Fn of Fx based on type vectors, where X is a finite set of points in $P^n$.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.393-404
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• 2004

We introduce several algorithms for adding up Artinian O-sequences to obtain the maximal possible Betti numbers among all minimal free resolutions with the given Hilbert function. Moreover, we give open questions based on the outputs using those algorithms.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.1077-1097
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• 2019

Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

• Journal of applied mathematics & informatics
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• v.5 no.2
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• pp.547-551
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• 1998

We give some examples of extremal minimal free reso-lutions of Gorenstein ideals of codimension 4.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.605-614
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• 2005

We show that an Artinian O-sequence $h_0,h_1,{\cdots},h_{d-1},h_d\;=\;h_{d-1},h_{d+l}\;>\;h_d$ of codimension 3 is not level when $h_{d-1}\;=\;h_d\;=\;d + i\;and\;h{d+1}\;=\;d+(i+1)\;for\;i\;=\;1,\;2,\;and\;3$, which is a partial answer to the question in [9]. We also introduce an algorithm for finding noncancelable Betti numbers of minimal free resolutions of all possible Artinian O-sequences based on the theorem of Froberg and Laksov in [2].

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.201-210
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• 2004

We examine a k-configuration X in P$^2$ or P$^3$ whose minimal free resolution has a non-cancelable Betti number in the last free module. We also find partial answers to the question: which Artinian O-sequences are level or not?

• Communications of the Korean Mathematical Society
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• v.8 no.4
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• pp.561-567
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• 1993

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.507-512
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• 2005

We find a condition that a graded Artinian O-sequence of codimension 4 is not level.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.405-417
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• 2012

We find the Hilbert function and the minimal free resolution of a star-configuration in $\mathbb{P}^n$. The conditions are provided under which the Hilbert function of a star-configuration in $\mathbb{P}^2$ is generic or non-generic We also prove that if $\mathbb{X}$ and $\mathbb{Y}$ are linear star-configurations in $\mathbb{P}^2$ of types t and s, respectively, with $s{\geq}t{\geq}3$, then the Artinian k-algebra $R/(I_{\mathbb{X}}+I_{\mathbb{Y})$ has the weak Lefschetz property.

• 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.