• Journal of the Chungcheong Mathematical Society
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• v.32 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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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.

• Honam Mathematical Journal
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• v.42 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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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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• v.55 no.3
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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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• v.37 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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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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• v.28 no.2
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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.

• Honam Mathematical Journal
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• v.41 no.4
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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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• v.43 no.3
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• pp.213-217
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• 2000

In order to investigate the removal effect of cadmium by polyphenol compound extracted from persimmon leaves(Diospyros kaki folium), animal test was done. Two fractions such as F-1 and F-2 from persimmon leaves were compared with their safety and function. The food intake of group F-1 and F-2 considerably decreased within 1% level. Cadmium addition influenced to rat growth(a tested animal), but food efficiency ratio(FER) wasn't shown any considerable difference in F-1 and F-2, as compared to the control. Cadmium content of liver, kidney and femur considerably decreased within 1% level in F-1 and F-2, compared to the control, cadmium content of liver decreased 25% in F-1, 28% in F-2 3150 decreased 22% md 25% in kidney. In femur, also decreased 53% in F-1 and 59% in F-2 respectively. The test of cadmium content in feces indicate that the content considerably increased within 1% level in both group F-1 and F-2, as compared to the control(42% in F-1, 54% in F-2). As shown the above results, have seen the removal effect of cadmium by polyphenol compound.