• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.319-329
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• 2017

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.615-623
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• 2013

In this paper we establish a central limit theorem for weighted sums of $Y_n={\sum_{i=1}^{n}}a_n,_iX_i$, where $\{a_{n,i},\;n{\in}N,\;1{\leq}i{\leq}n\}$ is an array of nonnegative numbers such that ${\sup}_{n{\geq}1}{\sum_{i=1}^{n}}a_{n,i}^2$ < ${\infty}$, ${\max}_{1{\leq}i{\leq}n}a_{n,i}{\rightarrow}0$ and $\{X_i,\;i{\in}N\}$ is a sequence of linear negatively quadrant dependent random variables with $EX_i=0$ and $EX_i^2$ < ${\infty}$. Using this result we will obtain a central limit theorem for partial sums of linear processes.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.573-580
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• 2006

In this paper, we obtain sufficient conditions for the oscillation of every solution of the difference equation $$x_{n+1}-x_n+\sum_{i=1}^{m}p_ix_{n-k_i}+qx_{n-z}=0,\;n=0,1,2,{\cdots},$$ where $p_i{\in}\mathbb{R}$, $k_i{\in}\mathbb{Z}$ for $i=1,2,{\cdots},m$ and $z{\in}\{-1,0\}$. Furthermore, we obtain sufficient conditions for the oscillation of all solutions of the equation $${\Delta}^rx_n+\sum_{i=1}^{m}p_ix_{n-k_i}=0,\;n=0,1,2,{\cdots},$$ where $p_i{\in}\mathbb{R}$, $k_i{\in}\mathbb{Z}$ for $i=1,2,{\cdots},m$. The results are given terms of the $p_i$ and the $k_i$ for each $i=1,2,{\cdots},m$.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.695-701
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• 2016

Let I and J be two ideals of a commutative Noetherian ring R and M, N be two non-zero finitely generated R-modules. Let t be a non-negative integer such that $H^i_{I,J}(N)$ is (I, J)-minimax for all i < t. It is shown that the generalized local cohomology module $H^i_{I,J}(M,N)$ is (I, J)-Cofinite minimax for all i < t. Also, we prove that the R-module $Ext^j_R(R/I,H^i_{I,J}(N))$ is finitely generated for all $i{\leq}t$ and j = 0, 1.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.471-477
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• 2019

Let (R, m) be a commutative Noetherian ring, I an ideal of R and M a non-zero finitely generated R-module. We show that if M and $H_0(I,M)$ are aCM R-modules and $I=(x_1,{\cdots},x_{n+1})$ such that $x_1,{\cdots},x_n$ is an M-regular sequence, then $H_i(I,M)$ is an aCM R-module for all i. Moreover, we prove that if R and $H_i(I,R)$ are aCM for all i, then R/(0 : I) is aCM. In addition, we prove that if R is aCM and $x_1,{\cdots},x_n$ is an aCM d-sequence, then depth $H_i(x_1,{\cdots},x_n;R){\geq}i-1$ for all i.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.823-830
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• 2021

Let {R_i}_i∈I be an infinite family of rings and R = ∏_i∈I R_i their product. In this paper, we investigate the prime spectrum of R by 𝓕-limits. Special attention is paid to relationship between the elements of Spec(R_i) and the elements of Spec(∏_i∈I R_i) use 𝓕-lim, also we give a new condition so that ∏_i∈I R_i is a zero dimensional ring.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.327-339
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• 2022

In this paper, we introduce arithmetic ${\mathcal{I}}$-statistically convergent sequence space $A{\mathcal{I}}SC$, ${\mathcal{I}}$-lacunary arithmetic statistically convergent sequence space $A{\mathcal{I}}SC_{\theta}$, strongly ${\mathcal{I}}$-lacunary arithmetic convergent sequence space $AN_{\theta}[{\mathcal{I}}]$ and prove some inclusion relations between these spaces. Futhermore, we give ${\mathcal{I}}$-lacunary arithmetic statistical continuity. Finally, we define ${\mathcal{I}}$-Cesàro arithmetic summability, strongly ${\mathcal{I}}$-Cesàro arithmetic summability. Also, we investigate the relationship between the concepts of strongly ${\mathcal{I}}$-Cesàro arithmetic summability, strongly ${\mathcal{I}}$-lacunary arithmetic summability and arithmetic ${\mathcal{I}}$ -statistically convergence.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.273-280
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• 2024

Let I be an ideal of a commutative Noetherian semi-local ring R and M be an R-module. It is shown that if dim M ≤ 2 and Supp_R M ⊆ V (I), then M is I-weakly cofinite if (and only if) the R-modules Hom_R(R/I, M) and Ext¹_R(R/I, M) are weakly Laskerian. As a consequence of this result, it is shown that the category of all I-weakly cofinite modules X with dim X ≤ 2, forms an Abelian subcategory of the category of all R-modules. Finally, it is shown that if dim R/I ≤ 2, then for each pair of finitely generated R-modules M and N and each pair of the integers i, j ≥ 0, the R-modules Tor^R_i(N, H^j_I(M)) and Extⁱ_R(N, H^j_I(M)) are I-weakly cofinite.

• Journal of Plant Biotechnology
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• v.31 no.1
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• pp.19-23
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• 2004

This study was conducted to observe the morphological characteristics and to analyse genetic relationship using RAPD markers in 9 species of Iris. The period of flowering was shown very broad among 9 species of Iris, from April 22 to June 9. The flower stalk was fluctuated from 9.0cm (I. rossii) to 104.0cm (I. ensata). I. ensata has the largest flower size and followed by I. laevigata, I. tectorum, I. sanguinea, I. pseudacorus, I. lactea, I. rossii, I. minutoaurea, I. odaesanensis. Among the 108 amplified bands by PCR with 8 random primers, 107 showed polymorphism and only one showed monomorphism among 9 species of Iris. The average dissimilarity coefficient among 9 species of Iris was 0.252. The range of dissimilarity coefficient was shown as from 0.095 to 0.609. Nine species were divided into three groups. The first group was I. tectorum, and the second group was I. lactea, I. laevigata, I. pseudacorus, I. odaesanensis. I. minutoaurea, I. rossii, I. sanguinea, I. ensata were belong to group three.

• Analytical Science and Technology
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• v.11 no.4
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• pp.248-253
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• 1998

The adsorption and separation behaviors of alkali, alkaline earth and transition metal ions using a poly-dibenzo-18-crown-6 were investigated in aqueous solution. The adsorption degree(E) and distribution ratio(D) of alkali, alkaline earth metal ions were Li(I)$t_R$) of metal ions were affected by the adsorption degree(E) and distribution ratio(D). This results showed good separation efficiency of K(I), Sr(II), Ag(I) and Pb(II) from the mixed metal solution.