• 대한수학회보
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• 제30권1호
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• pp.53-59
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• 1993

In this paper, we want to find a subgroup HJ(f, $x_{0}$, G) of the extended Jiang subgroup of a transformation group which is contained in Z( $f_{\sigma}$(.sigma.(X, $x_{0}$, G)), .sigma.(X, f( $x_{0}$), G)) and is an extension of the Jiang subgroup J(f, $x_{0}$). This is, if the acting group G is the trivial group {1x}, then this is the Jiang's results.ults..ults.

• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.615-631
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• 2023

A power cordial labeling of a graph G = (V (G), E(G)) is a bijection f : V (G) → {1, 2, ..., |V (G)|} such that an edge e = uv is assigned the label 1 if f(u) = (f(v))ⁿ or f(v) = (f(u))ⁿ, For some n ∈ ℕ ∪ {0} and the label 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In this paper, we investigate power cordial labeling for helm graph, flower graph, gear graph, fan graph and jewel graph as well as larger graphs obtained from star and bistar using graph operations.

• Journal of applied mathematics & informatics
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• 제42권2호
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• pp.445-456
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• 2024

A power cordial labeling of a graph G = (V (G), E(G)) is a bijection f : V (G) → {1, 2, ..., |V (G)|} such that an edge e = uv is assigned the label 1 if f(u) = (f(v))ⁿ or f(v) = (f(u))ⁿ, for some n ∈ ℕ ∪ {0} {0} and the label 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In this paper, we study power cordial labeling and investigate power cordial labeling for some standard graph families.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제3권2호
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• pp.163-171
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• 1996

Observing that a locally weakly Lindel$\"{o}$f space is a quasi-F space if and only if it has an F-base, we show that every dense weakly Lindel$\"{o}$f subspace of an almost-p-space is C-embedded, every locally weakly Lindel$\"{o}$f space with a cocompact F-base is a locally compact and quasi-F space and that if Y is a dense weakly Lindel$\"{o}$f subspace of X which has a cocompact F-base, then $\beta$Y and X are homeomorphic. We also show that for any a separating nest generated intersection ring F on a space X, there is a separating nest generated intersection ring g on $\phi_{Y}^{-1}$(X) such that QF(w(X, F)) and ($\phi_{Y}^{-1}$(X),g) are homeomorphic and $\phi_{Y}_{x}$(g$^#$)=F$^#$.

• Kyungpook Mathematical Journal
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• 제55권3호
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• pp.641-652
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• 2015

In this paper we study the problem of normal families of meromorphic functions concerning shared values. Let F be a family of meromorphic functions in the plane domain $D{\subseteq}{\mathbb{C}}$ and n, k be two positive integers such that $n{\geq}k+1$, and let a, b be two finite complex constants such that $a{\neq}0$. Suppose that (1) $f+a(f^{(k)})^n$ and $g+a(g^{(k)})^n$ share b in D for every pair of functions f, $g{\in}F$; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each $f{\in}F$ in D; (3) Zeros of $f^{(k)}(z)$ are not the b points of f(z) for each $f{\in}F$ in D. Then F is normal in D. And some examples are provided to show the result is sharp.

• 대한수학회보
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• 제26권2호
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• pp.169-173
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• 1989

Let V={(x,y,z):f=z$^{n}$ -npz+(n-1)q=0 for n .geq. 3} be a compled analytic subvariety of a polydisc in $C^{3}$ where p=p(x,y) and q=q(x,y) are holomorphic near (x,y)=(0,0) and f is an irreducible Weierstrass polynomial in z of multiplicity n. Suppose that V has an isolated singular point at the origin. Recall that the z-discriminant of f is D(f)=c(p$^{n}$ -q$^{n-1}$) for some number c. Suppose that D(f) is square-free. then we prove that by Theorem 2.1 .mu.(p$^{n}$ -q$^{n-1}$)=.mu.(f)-(n-1)+n(n-2)I(p,q)+1 where .mu.(f), .mu. p$^{n}$ -q$^{n-1}$are the corresponding Milnor numbers of f, p$^{n}$ -q$^{n-1}$, respectively and I(p,q) is the intersection number of p and q at the origin. By one of applications suppose that W$_{t}$ ={(x,y,z):g$_{t}$ =z$^{n}$ -np$_{t}$ $^{n-1}$z+(n-1)q$_{t}$ $^{n-1}$=0} is a smooth family of complex analytic varieties near t=0 each of which has an isolated singularity at the origin, satisfying that the z-discriminant of g$_{t}$ , that is, D(g$_{t}$ ) is square-free. If .mu.(g$_{t}$ ) are constant near t=0, then we prove that the family of plane curves, D(g$_{t}$ ) are equisingular and also D(f$_{t}$ ) are equisingular near t=0 where f$_{t}$ =z$^{n}$ -np$_{t}$ z+(n-1)q$_{t}$ =0.}$ =0.

• 한국수학교육학회지시리즈A:수학교육
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• 제21권3호
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• pp.7-11
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• 1983

A measurable function f defined on a measurable subset A of the real line R is called pth power summable on A if │f│$^{p}$ is integrable on A and the set of all pth power summable functions on A is denoted by L$^{p}$ (A). For each member f in L$^{p}$ (A), we define ∥f∥$_{p}$ =(equation omitted) For real numbers p and q where (equation omitted) and (equation omitted), we discuss the Holder's inequality ∥fg∥$_1$<∥f∥$_{p}$ ∥g∥$_{q}$ , f$\in$L$^{p}$ (A), g$\in$L$^{q}$ (A) and the Minkowski inequality ∥＋g∥$_{p}$ <∥f∥$_{p}$ ＋∥g∥$_{p}$ , f,g$\in$L$^{p}$ (A). In this paper also discuss that L$_{p}$ (A) becomes a metric space with the metric $\rho$ : L$^{p}$ (A) $\times$L$^{p}$ (A) longrightarrow R where $\rho$(f,g)=∥f-g∥$_{p}$ , f,g$\in$L$^{p}$ (A). Then, in this paper prove the Riesz-Fischer theorem, i.e., the space L$^{p}$ (A) is complete and that the space L$^{p}$ (A) is separable.

• 상하수도학회지
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• 제26권6호
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• pp.825-831
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• 2012

The retrofitting of a thickening unit process is widely considered in municipal wastewater treatment plants in Korea to enhance the anaerobic digestion efficiency. The authors examined the effect of feed concentration (2-34.1 g VS/L) and feed to microorganism (F/M) ratio (0.50-1.35 g VS/g VS) on anaerobic batch digestion of sewage sludge. Methane yield over 90 mL $CH_4/g$ $VS_{feed}$ was found at a feed concentration in the range of 12-26 g VS/L and a F/M ratio below 0.6 g VS/g VS. A high F/M ratio decreased methane yield and rate with oragnic acid accumulation. As sudden increase of sewage sludge concentration prior to anaerobic digestion would jeopardize the digester performance due to the rasied F/M ratio, gradual increase of the sludge feed concentration or an additional biomass retention in the digester is recommended.

• 대한수학회보
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• 제56권1호
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• pp.45-56
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• 2019

The aim of this paper is to investigate the problems of meromorphic functions sharing values concerning periodicities and shifts. In this paper we prove the following result: Let f(z) and g(z) be two nonconstant entire functions, let $c{\in}{\mathbb{C}}{\setminus}\{0\}$, and let $a_1$, $a_2$ be two distinct finite complex numbers. Suppose that ${\mu}(f){\neq}1$, ${\rho}_2(f)<1$, and f(z) = f(z+c) for all $z{\in}{\mathbb{C}}$. If f(z) and g(z) share $a_1$ CM, $a_2$ IM, then $f(z){\equiv}g(z)$. Moreover, examples are given to show that all the conditions are necessary.

• 대한수학회지
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• 제52권5호
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• pp.991-1001
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• 2015

In this work we obtain conditions for nonspecial line bundles on general ${\kappa}$-gonal curves failing to be normally generated. Let L be a nonspecial very ample line bundle on a general ${\kappa}$-gonal curve X with ${\kappa}{\geq}4$ and $deg\mathcal{L}{\geq}{\frac{3}{2}}g+{\frac{g-2}{{\kappa}}}+1$. If L fails to be normally generated, then L is isomorphic to $\mathcal{K}_X-(ng^1_{\kappa}+B)+R$ for some $n{\geq}1$, B and R satisfying (1) $h^0(R)=h^0(B)=1$, (2) $n+3{\leq}degR{\leq}2n+2$, (3) $deg(R{\cap}F){\leq}1$ for any $F{\in}g^1_k $. Its converse also holds under some additional restrictions. As a corollary, a very ample line bundle $\mathcal{L}{\simeq}\mathcal{K}_X-g^0_d+{\xi}^0_e$ is normally generated if $g^0_d{\in}X^{(d)}$ and ${\xi}^0_e{\in}X^{(e)}$ satisfy $d{\leq}{\frac{g}{2}}-{\frac{g-2}{\kappa}}-3$, supp$(g^0_d{\cap}{\xi}^0_e)={\phi}$ and deg$(g^0_d{\cap}F){\leq}{\kappa}-2$ for any $F{\in}g^1_k$.