• Kyungpook Mathematical Journal
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• 제48권4호
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• pp.545-551
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• 2008

Necessary and sufficient conditions for a radical class of rings to satisfy the polynomial equation $\rho$(R[x]) = ($\rho$(R))[x] have been investigated. The interrelationsh of polynomial equation, Amitsur property and polynomial extensibility is given. It has been shown that complete analogy of R.E. Propes result for radicals of matrix rings is not possible for polynomial rings.

• Nonlinear Functional Analysis and Applications
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• 제28권3호
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• pp.801-812
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• 2023

In this paper, we investigate the superstability for the p-power-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)$$ from an approximation of the p-power-radical functional equation: $$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 and f, g, h are complex valued functions. Furthermore, the obtained results are extended to Banach algebras.

• Korean Journal of Mathematics
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• 제29권4호
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• pp.765-774
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• 2021

In this paper, we solve and investigate the superstability of the p-radical functional equations $$f(\sqrt[p]{x^p+y^p})-f(\sqrt[p]{x^p-y^p})={\lambda}f(x)g(y),\\f(\sqrt[p]{x^p+y^p})-f(\sqrt[p]{x^p-y^p})={\lambda}g(x)f(y),$$ which is related to the trigonometric(Kim's type) functional equations, where p is an odd positive integer and f is a complex valued function. Furthermore, the results are extended to Banach algebras.

• 대한수학회보
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• 제58권2호
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• pp.269-275
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• 2021

Let ℝ be the set of real numbers, f, g : ℝ → ℝ and �� ≥ 0. In this paper, we consider the stability of partially pexiderized exponential-radical functional equation $$f({\sqrt[n]{x^N+y^N}})=f(x)g(y)$$ for all x, y ∈ ℝ, i.e., we investigate the functional inequality $$\|f({\sqrt[n]{x^N+y^N}})-f(x)g(y)\|{\leq}{\epsilon}$$ for all x, y ∈ ℝ.

• 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.

• Journal of Photoscience
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• 제8권1호
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• pp.33-37
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• 2001

Picosecond absorption spectroscopy has been employed in the study of the solvent dynamics of 1, 2, 4, 5-tetracyanobenzene/biphenyl derivative radical ion pairs, and the resulting rates of radical ion pair separation are faster in acetonitrile than in dichloromethane. In an effort to account quantitatively for such solvent effect on the rate of radical ion pair separation, an equation for the rate of radical ion pair separation is introduced, in which the rate depends exponentially on the electrostatic interaction energy in the radical ion pair. In our analysis of the types of electrostatic interaction energy based on the conducting spheres in dielectric continuum was chosen, and the rate equation employing this electrostatic energy provided information on the distance on the distance of radical ion pair separation and solvation energy of the radical ion pair, thereby providing quantitative explanation for the observed solvent effect on the rate of radical ion pair sepaaration.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제28권4호
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• pp.387-398
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• 2021

In this paper, we will find solutions and investigate the superstability bounded by constant for the p-radical functional equations as follows: $f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=\;\{(i)\;f(x)f(y),\\(ii)\;g(x)f(y),\\(iii)\;f(x)g(y),\\(iv)\;g(x)g(y).$ with respect to the sine functional equation, where p is an odd positive integer and f is a complex valued function. Furthermore, the results are extended to Banach algebra.

• 대한기계학회논문집B
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• 제27권9호
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• pp.1309-1316
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• 2003

A premixed-compression-ignition engine has been studied to improve the efficiency and to decrease exhaust emissions. However those systems have some difficulties for controlling combustion process. Radical is an activated chemical species formed by a chemical chain reaction between reactant and product. When the chain reactions occur, the energy bond of species is broken easily by the released radicals. The combustion chamber of the premixed-compression-ingnition engine is consist of a main chamber with lean premixture and a subchamber with rich premixture. Those are connected by narrow cylinderical connections. With ignition start in the subchamber, many different kinds of radical is jetted into the main chamber. The premixed gas in main chamber is quickly burned up by the radical ignition in multi-pionts. In this paper, the combustion phenomena in a constant volume combustor having a radical injector are numerically analyzed. The some constants in the reaction rate equation are adjusted by the experimental results tested in the same geometrical chamber. The code is applied on the two combustors in a wide range of equivalence ratio. The results show that the burning time is much shorter in the combustor having radical injector.

• 대한수학회논문집
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• 제36권2호
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• pp.343-359
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• 2021

The aim of this paper is to obtain the general solution of the 2-variable radical functional equations $f({\sqrt[k]{x^k+z^k}},\;{\sqrt[{\ell}]{y^{\ell}+w^{\ell}}})=f(x,y)+f(z,w)$, x, y, z, w ∈ ℝ, for f a mapping from the set of all real numbers ℝ into a vector space, where k and ℓ are fixed positive integers. Also using the fixed point result of Brzdęk and Ciepliński [11, Theorem 1] in (2, 𝛽)-Banach spaces, we prove the generalized hyperstability results of the 2-variable radical functional equations. In the end of this paper we derive some consequences from our main results.

• 대한화학회지
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• 제27권6호
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• pp.391-398
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• 1983

-butoxyl, t-butyl라디칼에 의한 치환체-톨루엔의 수소추출 반응에 대하여 CNDO/2 방법을 써서 분자궤도론적으로 고찰하였다. t-Butoxyl 라디칼의 치환체-톨루엔에 대한 수소추출반응이 Hammett식에 의하여 음의 ${\rho}$값을 나타내는 것은 t-butoxyl 라디칼이 낮은 SOMO에너지를 갖는 친전자성 라디칼의 성질이 크므로 치환체-톨루엔의 HOMO와 작용하기 때문이다. 반면, t-butyl 라디칼은 높은 SOMO에너지를 갖는 친핵성라디칼의 성질이 증가하므로 치환체-톨루엔의 LUMO와의 작용이 커져 양의 ${\rho}$값을 가진다.