• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.83-94
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• 2016

Let M and N be modules over a ring R. The purpose of this paper is to study modules M, N for which the bi-module [M, N] is regular or pi. It is proved that the bi-module [M, N] is regular if and only if a module N is semi M-projective and $Im({\alpha}){\subseteq}^{\oplus}N$ for all ${\alpha}{\in}[M,N]$, if and only if a module M is semi N-injective and $Ker({\alpha}){\subseteq}^{\oplus}N$ for all ${\alpha}{\in}[M,N]$. Also, it is proved that the bi-module [M, N] is pi if and only if a module N is direct M-projective and for any ${\alpha}{\in}[M,N]$ there exists ${\beta}{\in}[M,N]$ such that $Im({\alpha}{\beta}){\subseteq}^{\oplus}N$, if and only if a module M is direct N-injective and for any ${\alpha}{\in}[M,N]$ there exists ${\beta}{\in}[M,N]$ such that $Ker({\beta}{\alpha}){\subseteq}^{\oplus}M$. The relationship between the Jacobson radical and the (co)singular ideal of [M, N] is described.

• 대한수학회보
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• 제61권1호
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• pp.1-12
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• 2024

In this paper, we introduce the notion of Gorenstein (m, n)-flat modules as an extension of (m, n)-flat left R-modules over a ring R, where m and n are two fixed positive integers. We demonstrate that the class of all Gorenstein (m, n)-flat modules forms a Kaplansky class and establish that (𝓖𝓕_m,n(R),𝓖𝓒_m,n(R)) constitutes a hereditary perfect cotorsion pair (where 𝓖𝓕_m,n(R) denotes the class of Gorenstein (m, n)-flat modules and 𝓖𝓒_m,n(R) refers to the class of Gorenstein (m, n)-cotorsion modules) over slightly (m, n)-coherent rings.

• 대한수학회보
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• 제36권4호
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• pp.799-808
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• 1999

We investigate when the .product of two smooth manifolds admits a weakly Lagrangian embedding. Assume M, N are oriented smooth manifolds of dimension m and n,. respectively, which admit weakly Lagrangian immersions into $C^m$ and $C^n$. If m and n are odd, then $M\;{\times}\;N$ admits a weakly Lagrangian embedding into $C^{m+n}$ In the case when m is odd and n is even, we assume further that $\chi$(N) is an even integer. Then $M\;{\times}\;N$ admits a weakly Lagrangian embedding into $C^{m+n}$. As a corollary, we obtain the result that $S^n_1\;{\times}\;S^n_2\;{\times}\;...{\times}\;S^n_k$, $\kappa$>1, admits a weakly Lagrang.ian embedding into $C^n_1+^n_2+...+^n_k$ if and only if some ni is odd.

• 대한수학회보
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• 제59권4호
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• pp.917-928
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• 2022

Let R be a commutative Noetherian ring and I an ideal of R. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let M be a finitely generated R-module and N a representable R-module. We introduce the notions of the representation dimension r^I(M, N) and artinianness dimension a^I(M, N) of M, N with respect to I by r^I(M, N) = inf{i ∈ ℕ₀ : H^I_i(M, N) is not representable} and a^I(M, N) = inf{i ∈ ℕ₀ : H^I_i(M, N) is not artinian} and we show that a^I(M, N) = r^I(M, N) = inf{r^IR_𝔭 (M_𝔭,𝔭N) : 𝔭 ∈ Spec(R)} ≥ inf{a^IR_𝔭 (M_𝔭,𝔭N) : 𝔭 ∈ Spec(R)}. Also, in the case where R is semi-local and N a semi discrete linearly compact R-module such that N/∩_t>0I^tN is artinian we prove that inf{i : H^I_i(M, N) is not minimax}=inf{r^IR_𝔭 (M_𝔭,𝔭N) : 𝔭 ∈ Spec(R)＼Max(R)}.

• 대한수학회보
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• 제61권3호
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• pp.717-734
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• 2024

Let R be a commutative ring with identity and m, n be positive integers. In this paper, we introduce the class of weakly (m, n)-prime ideals generalizing (m, n)-prime and weakly (m, n)-closed ideals. A proper ideal I of R is called weakly (m, n)-prime if for a, b ∈ R, 0 ≠ a^mb ∈ I implies either aⁿ ∈ I or b ∈ I. We justify several properties and characterizations of weakly (m, n)-prime ideals with many supporting examples. Furthermore, we investigate weakly (m, n)-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behaviour of this class of ideals in idealization and amalgamated rings.

• Journal of applied mathematics & informatics
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• 제42권2호
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• pp.353-366
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• 2024

A total coloring of a graph G is an assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices and edges of G are assigned the same colors. In this paper, we have discussed the total coloring of M(T_n), M(D_n), M(DT_n), M(AT_n), M(DA(T_n)), M(Q_n), M(AQ_n) and also obtained the total chromatic number of M(T_n), M(D_n), M(DT_n), M(AT_n), M(DA(T_n)), M(Q_n), M(AQ_n).

• 한국컴퓨터정보학회논문지
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• 제18권8호
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• pp.87-93
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• 2013

최단 보폭-최장 보폭 알고리즘은 n을 $m={\lceil}\sqrt{n}{\rceil}$개의 원소를 가진 m개의 블록으로 분할하고 첫 번째 블록의 m개에 대해 $a^x$ (mod n) 값을 저장한다. 다음으로 m개의 블록에 대한 mod n을 계산하여 첫 번째 블록의 원소 값을 검색하여 일치하는 블록을 찾는 방법이다. 본 논문에서는 첫 번째로, $a^{{\phi}(n)/2}{\equiv}1(mod\;n)$과 $a^x(mod\;n){\equiv}a^{{\phi}(n)+x}$ (mod n)의 특징을 적용하여 m개의 원소를 가진 ${\lceil}m/2{\rceil}$개의 블록으로 분할하는 방법을 적용하여 최장보폭의 수행횟수를 50% 감소시켰다. 두 번째로, ${\lceil}m/2{\rceil}$개의 최단 보폭을 먼저 수행하여 저장하고, 첫 번째 블록의 m개 원소를 수행하는 최단 보폭을 수행하는 방법으로 최단 보폭-최장 보폭 알고리즘을 역으로 수행하는 방법을 제안하였다. 이 알고리즘은 최단 보폭-최장 보폭 알고리즘의 m개 저장과 검색을 ${\lceil}m/2{\rceil}$개로 50% 감소시키는 특징이 있다.

• 대한수학회논문집
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• 제17권4호
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• pp.719-729
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• 2002

Consider the random n $\times$ m assignment problem with m $\geq$ $_{i,j}$ Let $u_{i,j}$ be iid uniform random variables on [0, 1] and exponential random variables with mean 1, respectively, and let $U_{n, m}$ and $T_{n, m}$ denote the optimal assignment costs corresponding to $u_{i, j}$ and $t_{i, j}$. In this paper we first give a comparison result about the discrepancy E $T_{n, m}$ -E $U_{n, m}$. Using this comparison result with a known lower bound for Var( $T_{n, m}$) we obtains a lower bound for Var( $U_{n, m}$). Finally, we study the way that E $U_{n, m}$ and E $T_{n, m}$ vary as m does. It turns out that only when m - n is large enough, the cost decreases significantly.tly.

• 대한수학회지
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• 제55권5호
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• pp.1031-1043
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• 2018

Let R be a commutative ring with $1{\neq}0$, I a proper ideal of R, and m and n positive integers. In this paper, we define I to be a weakly (m, n)-closed ideal if $0{\neq}x^m\;{\in}I$ for $x{\in}R$ implies $x^n{\in}I$, and R to be an (m, n)-von Neumann regular ring if for every $x{\in}R$, there is an $r{\in}R$ such that $x^mr=x^n$. A number of results concerning weakly(m, n)-closed ideals and (m, n)-von Neumann regular rings are given.

• 대한수학회보
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• 제35권4호
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• pp.809-817
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• 1998

We investigate when the product of two smooth manifolds admits a weakly Lagrangian embedding. Prove that, if $M^m$ and $N^n$ are smooth manifolds such that M admits a weakly Lagrangian embedding into ${\mathbb}C^m$ whose normal bundle has a nowhere vanishing section and N admits a weakly Lagrangian immersion into ${\mathbb}C^n$, then $M \times N$ admits a weakly Lagrangian embedding into ${\mathbb}C^{m+n}$. As a corollary, we obtain that $S^m {\times} S^n$ admits a weakly Lagrangian embedding into ${\mathbb}C^{m+n}$ if n=1,3. We investigate the problem of whether $S^m{\times}S^n$ in general admits a weakly Lagrangian embedding into ${\mathbb} C^{m+n}$.