• 대한수학회논문집
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• 제19권4호
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• pp.701-713
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• 2004

We obtain the Hyers-Ulam-Rassias stability of a betatype functional equation $f(\varphi(x),\phi(y))$ = $ \psi(x,y)f(x,y)+ \lambda(x,y)$ with a restricted domain and the stability in the sense of R. Ger of the equation $f(\varphi(x),\phi(y))$ = $ \psi(x,y)f(x,y)$ with a restricted domain in the following settings: $g(\varphi(x),\phi(y))-\psi(x,y)g(s,y)-\lambda(x,y)$\mid$\leq\varepsilon(x,y)$ and $\frac{g(\varphi(x),\phi(y))}{\psi(x,y),g(x,y)}-1 $\mid$ \leq\epsilon(x,y)$.

• KIEE International Transactions on Electrophysics and Applications
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• 제4C권2호
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• pp.45-50
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• 2004

The field utilization factor (equation omitted) (the mean electric field / the maximum electric field) of standard sphere gaps was calculated by the charge simulation method, taking into account the ground plane and shanks. n changes mainly with g/r and slightly with 1$_1$, 1$_2$ and 1, where D=2r is the sphere diameter, g is the gap length, 1$_1$ and 1$_2$, respectively, are the lengths of the upper and lower shank, and t is the shank diameter. Generally, (equation omitted) increases as 1$_1$,1$_2$ and t each becomes larger. IEC standard 60052(2002) limits t$\leq$0.2D 1$_1$$\geq$1D and prescribes A=1$_2$＋D＋g where A is the height of the spark point on the upper sphere. Therefore, (equation omitted) is the largest when A=9D and the smallest when A=3D. The simple equation of a straight line, (equation omitted)=1- (g/3r), can generally be used as a representative value of (equation omitted) for a wide variety of sphere diameters that are permitted by the IEC standard. The maximum electric field E$_{m}$ at sparkover of standard air gaps has also been calculated by the relation E$_{m}$=V/(equation omitted)g). E$_{m}$ describes a U-curve for g/r, up to the sphere diameter of 1 m. Moreover, for 1.5-m and 2-m diameters and especially .for negative polarity, sparkover voltages have been calculated by integration of the ionization index.index.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1421-1433
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• 2011

In this paper, using a fixed point approach, the generalized Hyeres-Ulam stability of the following quadratic functional equation $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=3(f(x)+f(y)+f(z))$ will be studied, where f is a function from abelian group G into a quasi-${\beta}$-normed space and ${\sigma}$ is an involution on the group G. Next, we consider its pexiderized equation of the form $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=g(x)+g(y)+g(z)$ and its generalized Hyeres-Ulam stability.

• Bulletin of the Korean Chemical Society
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• 제22권7호
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• pp.732-738
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• 2001

The cross second-derivative of the activation energy,${\theta}$G${\neq}$ , with respect to the two component thermodynamic barriers, ${\theta}$G˚X and ${\theta}$G$^{\circ}C$Y, can be given in terms of a cross-interaction constant (CIC), $\betaXY(\rhoXY)$, and also in terms of the intrinsic barrier,${\theta}$G${\neq}$ , with a simple relationship between the two: $\betaXY$ = $-1}(6${\theta}$G${\neq}$).$ This equation shows that the distance between the two reactants in the adduct (TS, intermediate, or product) is inversely related to the intrinsic barrier. An important corollary is that the Ritchie N+ equation holds (for which $\betaXY$ = 0) for the reactions with high intrinsic barrier. Various experimental and theoretical examples are presented to show the validity of the relationship, and the mechanistic implications are discussed.

• 호남수학학술지
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• 제31권3호
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• pp.451-462
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• 2009

In this paper we prove the superstability of a generalized exponential functional equation $f(x+y)=a^{2xy-1}g(x)f(y)$. It is a generalization of the superstability theorem for the exponential functional equation proved by Baker. Also we investigate the stability of this functional equation in the following form : ${\frac{1}{1+{\delta}}}{\leq}{\frac{f(x+y)}{a^{2xy-1}g(x)f(y)}}{\leq}1+{\delta}$.

• 충청수학회지
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• 제25권1호
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• pp.1-18
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• 2012

The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation $f(\frac{x+y}{2})^{2}-f(\frac{x+{\sigma}y}{2})^{2}={\lambda}g(x)h(y),\;{\lambda}:\;constant$ which is bounded by the unknown functions ${\varphi}(x)$ or ${\varphi}(y)$. As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

• 한국수학교육학회지시리즈A:수학교육
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• 제10권2호
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• pp.4-8
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• 1972

We consider the class (equation omitted) of all k-degenerate graphs, for k a non-negative integer. The class (equation omitted) and (equation omitted) are exactly the classes of totally disconnected graphs and of forests, respectively; the classes (equation omitted) and (equation omitted) properly contain all outerplanar and planar graphs respectively. The advantage of this view point is that many of the known results for chromatic number and point arboricity have natural extensions, for all larger values of k. The purpose of this note is to show that a graph G is (P$^3$)-realizable if G is planar and 3-degenerate.

• 충청수학회지
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• 제35권4호
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• pp.315-327
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• 2022

In this paper, we investigate the transferred superstability for the p-radical sine functional equation $$f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=f(x)f(y)$$ from the p-radical functional equations: $$f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)g(y),\;\\f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)h(y),$$ where p is an odd positive integer, λ is a positive real number, and f is a complex valued function. Furthermore, the results are extended to Banach algebras. Therefore, the obtained result will be forced to the pre-results(p=1) for this type's equations, and will serve as a sample to apply it to the extension of the other known equations.

• 호남수학학술지
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• 제33권2호
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• pp.247-259
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• 2011

In this paper, the $(\frac{G'}{G})$-expansion method is used to construct new exact travelling wave solutions of some nonlinear evolution equations. The travelling wave solutions in general form are expressed by the hyperbolic functions, the trigonometric functions and the rational functions, as a result many previously known solitary waves are recovered as special cases. The $(\frac{G'}{G})$-expansion method is direct, concise, and effective, and can be applied to man other nonlinear evolution equations arising in mathematical physics.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.837-844
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• 2002

In this paper, we obtain the Hyers-Ulam Stability for the difference equations of the form f(x + 1) = $\Gamma$(x)f(x), which is the reciprocal functional equation of the double gamma function.