• 충청수학회지
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• 제32권3호
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• pp.337-347
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• 2019

In this paper, we investigate the generalized Hyers-Ulam stability of the quadratic and quartic type functional equations $$f(kx+y)+f(kx-y)-k^2f(x+y)-k^2f(x-y)-2f(kx)\\{\hfill{67}}+2k^2f(x)+2(k^2-1)f(y)=0,\\f(x+5y)-5f(x+4y)+10f(x+3y)-10f(x+2y)+5f(x+y)\\{\hfill{67}}-f(-x)=0,\\f(kx+y)+f(kx-y)-k^2f(x+y)-k^2f(x-y)\\{\hfill{67}}-{\frac{k^2(k^2-1)}{6}}[f(2x)-4f(x)]+2(k^2-1)f(y)=0$$ by using the fixed point theory in the sense of L. $C{\breve{a}}dariu$ and V. Radu.

• 대한수학회보
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• 제27권2호
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• pp.127-131
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• 1990

Let f:R.rarw.R be a continuous function on the real line R, and denote the n-th iterate of f by f$^{n}$ :f$^{1}$=f and f$^{n}$ =f.f$^{n-1}$ for n>1. A point x.mem.R is a periodic point of f of period k>0 if f$^{k}$ (x)=x but f$^{i}$ (x).neq.x for all 01, then it must also have a fixed point, by the intermediate Theorem. Also the question has an intriguing answer which was found by ths Russian mathematician Sharkovky [6] in 1964.

• 호남수학학술지
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• 제42권3호
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• pp.449-460
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• 2020

In this paper, we investigate the stability of the additive-cubic functional equations f(x+ky)+f(x-ky)-k² f(x+y)-k² f(x-y)+(k²-1)f(x) - (k²-1)f(-x) = 0, f(x+ky)-f(ky-x)-k² f(x+y)+k² f(y-x)+2(k²-1)f(x)= 0, f(kx+y)+f(kx-y)-kf(x+y)-kf(x-y)-2f(kx)+2kf(x)= 0 by using the fixed point theory in the sense of L. Cădariu and V. Radu.

• 대한수학회보
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• 제22권2호
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• pp.83-85
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• 1985

Let K be a real function field over a real closed field F. Then there exists an F-place .phi.:K.rarw.F.cup.{.inf.}. This is Lang's Existence of Rational Place Theorem (6). There is an equivalent version of Lang's Theorem in (4). That is, if K is a function field over a field F, then, for any ordering P$_{0}$ on F which extends to K, there exists an F-place .phi.: K.rarw.F'.cup.{.inf.} where F' is a real closure of (F, P$_{0}$). In [2], Knebusch pointed out the converse of the version of Lang's Theorem is also true. By a valuation theoretic approach to Lang's Theorem, we have found out the following generalization of Lang and Knebusch's Theorem. Let K be an arbitrary extension field of a field F. then an ordering P$_{0}$ on F can be extended to an ordering P on K if there exists an F-place of K into some real closed field R containing F. Of course R$^{2}$.cap.F=P$_{0}$. The restriction K being a function field of F is vanished, though the codomain of the F-place is slightly varied. Therefore our theorem is a generalization of Lang and Knebusch's theorem.

• 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.

• BMB Reports
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• 제37권2호
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• pp.199-205
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• 2004

The six lumbrokinase fractions (F1 to F6) with fibrinolytic activities were purified from earthworm Lumbricus rubellus lysates using the procedures of autolysis, ammonium sulfate fractionation, and column chromatography. The proteolytic activities on the casein substrate of the six iso-enzymes ranged from 11.3 to 167.5 unit/mg with the rank activity orders of F2 > F1 > F5 > F6 > F3 > F4. The fibrinolytic activities of the six fractions on the fibrin plates ranged from 20.8 to 207.2 unit/mg with rank orders of F6 > F2 > F5 > F3 > F1 > F4. The molecular weights of each iso-enzyme, as estimated by SDS-PAGE, were 24.6 (F1), 26.8 (F2), 28.2 (F3), 25.4 (F4), 33.1 (F5), and 33.0 kDa (F6), respectively. The plasminogen was activated into plasmin by the enzymes. The optimal temperature of the six iso-enzymes was $50^{\circ}C$, and the optimal pH ranged from pH 4-12. The four iso-enzymes (F1-F4) were completely inhibited by PMSF. The two enzymes (F5 and F6) were completely inhibited by aprotinin, TLCK, TPCK, SBTI, LBTI, and leupeptin. The N-terminal amino acid (aa) sequences of the first 20 to 22 residues of each fraction had high homology. All six isoenzymes had identical aa residues 2-3 and 13-15. The N-terminal 21-22 aa sequences of the F2, F3, and F4 isoenzymes were almost the same. The N-terminal aa sequences of F5 and F6 were identical.

• 대한수학회지
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• 제32권4호
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• pp.689-695
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• 1995

Let G be a finite group with k distinct conjugacy classes $C_1, C_2, \cdots, C_k$ and F an algebraically closed field such that char$(F){\dag}\left$\mid$ G \right$\mid$$. We denoted by $Irr_F$(G) the set of all irreducible F-characters of G and $Cf_F$(G) the set of all class functions of G into F. Then $Cf_F$(G) is a commutative F-algebra with an F-basis $Irr_F(G) = {\chi_1, \chi_2, \cdots, \chi_k}$.

• Korean Journal of Mathematics
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• 제28권2호
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• pp.159-168
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• 2020

In this paper, we investigate Hyers-Ulam-Rassias stability of a functional equation f(x + ky) + f(x - ky) - k²f(x + y) - k²f(x - y) + 2(k² - 1)f(x) + (k² + k³)f(y) + (k² - k³)f(-y) - 2f(ky) = 0.

• 호남수학학술지
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• 제41권4호
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• pp.813-821
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• 2019

In this paper, we investigate Hyers-Ulam-Rassias stability of a functional equation f(x + ky) + f(x - ky) - k²f(x + y) - k²f(x - y) + 2(k² - 1)f(x) + (k² + k)f(y) + (k² - k)f(-y) - 2f(ky) = 0.

• Applied Biological Chemistry
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• 제43권3호
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• pp.213-217
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• 2000

감잎(Diospyros kaki folium)으로부터 폴리페놀 화합물을 분리하여 중금속 제거효과의 동물실험을 통한 안전성 및 기능성을 검토하였다. 카드뮴을 단독 투여한 대조군에 대하여 카드뮴에 감잎을 병합 투여한 F-1군과 F-2군 모두 식이섭취량은 1% 수준에서 유의적으로 감소하였고, 체중증가량은 카드뮴이 식이에 첨가됨으로써 실험동물의 성장에 영향을 끼쳤으며, 식이효율은 카드뮴 단독 투여군에 대하여 F-1군과 F-2군 모두에서 유의적인 차이가 나타나지 않았다. 간, 신장 그리고 대퇴골의 카드뮴 함량을 분석한 결과, 카드뮴을 단독 투여한 대조군에 대하여 F-1군과 F-2군 모두 1% 수준에서 유의적으로 감소하였다. 간에서는 F-1군이 25%, F-2군은 28%씩 감소하였고, 신장에서는 F-1군이 22%, F-2군이 25%씩 감소하였고, 대퇴골에 있어서의 카드뮴은 F-1군이 53%, F-2군은 59% 감소하였다. 변의 카드뮴 함량을 분석한 결과 대조군에 대하여 F-1군과 F-2군 모두 1% 수준에서 유의적으로 증가하였으며, F-1군이 42% 그리고 F-2군은 54%증가하였다. 위 결과로 미루어 볼 때 폴리페놀 화합물의 카드뮴 제거효과를 관찰할 수 있었다.