• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.105-115
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• 2011

Let f be a function which assigns a positive integer f(v) to each vertex v $\in$ V (G), let r, s and t be non-negative integers. An f-coloring of G is an edge-coloring of G such that each vertex v $\in$ V (G) has at most f(v) incident edges colored with the same color. The minimum number of colors needed to f-color G is called the f-chromatic index of G and denoted by ${\chi}'_f$(G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) $\bigcup$ E(G) to the color set C = {0, 1, $\ldots$; k - 1} such that |c($v_i$) - c($v_j$ )| $\geq$ r for every two adjacent vertices $v_i$ and $v_j$, |c($e_i$ - c($e_j$)| $\geq$ s and ${\alpha}(v_i)$ $\leq$ f($v_i$) for all $v_i$ $\in$ V (G), ${\alpha}$ $\in$ C where ${\alpha}(v_i)$ denotes the number of ${\alpha}$-edges incident with the vertex $v_i$ and $e_i$, $e_j$ are edges which are incident with $v_i$ but colored with different colors, |c($e_i$)-c($v_j$)| $\geq$ t for all pairs of incident vertices and edges. The minimum k such that G has an [r, s, t; f]-coloring with k colors is defined as the [r, s, t; f]-chromatic number and denoted by ${\chi}_{r,s,t;f}$ (G). In this paper, we present some general bounds for [r, s, t; f]-coloring firstly. After that, we obtain some important properties under the restriction min{r, s, t} = 0 or min{r, s, t} = 1. Finally, we present some problems for further research.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.135-149
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• 2011

Using (r, s)-preopen sets [14] and pre-${\omega}_t$-closures [6], a new kind of covering property $P^t_{({\omega}_r,s)}$-closedness is introduced in a bitopological space and several characterizations via filter bases, nets and grills [30] along with various properties of such concept are investigated. Two new types of cluster sets, namely pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets and (r, s)t-${\theta}_f$-precluster sets of functions and multifunctions between two bitopological spaces are introduced. Several properties of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets are investigated and using the degeneracy of such cluster sets, some new characterizations of some separation axioms in topological spaces or in bitopological spaces are obtained. A sufficient condition for $P^t_{({\omega}_r,s)}$-closedness has also been established in terms of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.961-976
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• 2019

Let R be any commutative ring and S be any multiplicative closed set. We introduce an S-version of $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as R is noetherian if and only if $R_S$ is noetherian and every $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, we also investigate the $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension of R, and prove that $R_S$ is noetherian if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is zero; $R_S$ is coherent if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is at most one.

• The Mathematical Education
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• v.12 no.2
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• pp.5-12
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• 1974

단위원을 가지는 하환환에 있어서의 Prime Spectrum에 관하여 다음 세가지 사실을 증명하였다. 1. X를 환 R의 prime spectrum, C(X)를 X에서 정의되는 실연적함수의 환, X를 C(X)의 maximal spectrum이라 하면 X는 C(X)의 prime spectrum의 부분공간으로서의 한 T-space로 된다. N을 환 R의 nilradical이라 하면, R/N이 regula 이면 X와 X는 위상동형이다. 2. f: R$\longrightarrow$R'을 ring homomorphism, P를 R의 한 Prime ideal, $R_{p}$, R'$_{p}$를 각각 S=R-P 및 f(S)에 관한 분수환(ring of fraction)이라 하고, k(P)를 local ring $R_{p}$의 residue' field라 할 때, R'의 prime spectrum의 부분공간인 $f^{*-1}$(P)는 k(P)(equation omitted)$_{R}$R'의 prime spectrum과 위상동형이다. 단 f*는 f*(Q)=$f^{-1}$(Q)로서 정의되는 함수 s*:Spec(R')$\longrightarrow$Spec(R)이다. 3. X를 환 S의 prime spectrum, N을 R의 nilradical이라 할 때, 다음 네가지 사실은 동치이다. (1) R/N 은 regular 이다. (2) X는 Zarski topology에 관하여 Hausdorff 공간이다. (3) X에서의 Zarski topology와 constructible topology와는 일치한다. (4) R의 임의의 원소 f에 대하여 f를 포함하지 않는 R의 prime ideal 전체의 집합 $X_{f}$는 Zarski topology에 관하여 개집합인 동시에 폐집합이다.폐집합이다....

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.643-657
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• 2022

Let R be a ring and S a multiplicative subset of R. An R-module T is called u-S-torsion (u-always abbreviates uniformly) provided that sT = 0 for some s ∈ S. The notion of u-S-exact sequences is also introduced from the viewpoint of uniformity. An R-module F is called u-S-flat provided that the induced sequence 0 → A ⊗_R F → B ⊗_R F → C ⊗_R F → 0 is u-S-exact for any u-S-exact sequence 0 → A → B → C → 0. A ring R is called u-S-von Neumann regular provided there exists an element s ∈ S satisfying that for any a ∈ R there exists r ∈ R such that sα = rα². We obtain that a ring R is a u-S-von Neumann regular ring if and only if any R-module is u-S-flat. Several properties of u-S-flat modules and u-S-von Neumann regular rings are obtained.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.1-10
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• 2021

Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to ��, and defined by M ⋈�� JN := {(m, ��(m) + n) | m ∈ M and n ∈ JN}, where �� : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.617-622
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• 2021

Let R be a commutative ring with identity, R[X] the polynomial ring over R and S a multiplicative subset of R. Let U = {f ∈ R[X] | f is monic} and let N = {f ∈ R[X] | c(f) = R}. In this paper, we show that if S is an anti-Archimedean subset of R, then R is an S-Noetherian ring if and only if R[X]_U is an S-Noetherian ring, if and only if R[X]_N is an S-Noetherian ring. We also prove that if R is an integral domain and R[X]_U is an S-principal ideal domain, then R is an S-principal ideal domain.

• Korean Journal of Soil Science and Fertilizer
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• v.29 no.1
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• pp.53-59
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• 1996

This study was conducted to evaluate the effects of fowl dzopping. saw dust and rice hall on the soil microflora in vitro. The experiment was designed in seven treatments with the various organic materials and they were only soil (control). soil + fowl dropping (S+F), soil+fowl dropping+rice hull (S+F+R) soil+fowl dropping*saw dust (S+F+S). soil+chemical fertilizer (S+C.F), fowl dropping+rice hull (F+R) and fowl dropping+saw dust (F+S). All the samples of treatment were incubated in $28{\pm}2^{\circ}C$ condition and tested the activity of soil microflora for 84 days The activity of fungi, total bacteria, gram-negative bacteria and actinomycetes showed the highest values at, twenty-first day and the spore-forming bacteria was at forty-second day after incubation. The number of fungi and gram-negative bacteria showed the highest values in the treatment of F+S, the spore-forming bacteria and the actinomycetes were in the S+F+S. and the number of total bacteria was in the F+C.F., but in the treatment of F+R. all the microorganism except fungi showed the lowest values in their numbers. The composition ratio of dead bacteria was higher in the treatments of S+F+R and F+R than in those of others as 70% and 40% respectively. Actinomycetes isolated from the treatments of S+F and S+F+S were identified as Streptomyces sp.. Nocardia sp., Micromonospora sp. Actinomadura sp. and Saccharomonospora sp.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1541-1565
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• 2020

Let f be a meromorphic function of finite order ρ with few poles in the sense S_λ(r, f) := O(r^λ+ε) + S(r, f), where λ < ρ and ε ∈ (0, ρ - λ), and let g(f) := Σ^k_j=1b_j(z)f^(k_j)(z + c_j) be a linear delay-differential polynomial of f with small meromorphic coefficients b_j in the sense S_λ(r, f). The zero distribution of fⁿ(g(f))^s - b₀ is considered in this paper, where b₀ is a small function in the sense S_λ(r, f).

• IE interfaces
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• v.21 no.3
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• pp.254-261
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• 2008

Connected-(r,s)-out of-(m,n) : F system is an important topic in redundancy design of the complex system reliability and it's maintenance policy. Previous studies applied Monte Carlo simulation and genetic, simulated annealing algorithms to tackle the difficulty of maintenance policy problem. These algorithms suggested most suitable maintenance cycle to optimize maintenance pattern of connected-(r,s)-out of-(m,n) : F system. However, genetic algorithm is required long execution time relatively and simulated annealing has improved computational time but rather poor solutions. In this paper, we propose the ant colony optimization approach for connected-(r,s)-out of-(m,n) : F system that determines maintenance cycle and minimum unit cost. Computational results prove that ant colony optimization algorithm is superior to genetic algorithm, simulated annealing and tabu search in both execution time and quality of solution.