• Kyungpook Mathematical Journal
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• v.48 no.4
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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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• v.28 no.3
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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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• v.29 no.4
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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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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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.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.4
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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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• v.8 no.1
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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.

• The Pure and Applied Mathematics
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• v.28 no.4
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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.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.27 no.9
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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.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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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.

• Journal of the Korean Chemical Society
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• v.27 no.6
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• pp.391-398
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• 1983

The reaction for the hydrogen abstraction from substituted-toluenes by t-butoxyl and t-butyl radical have been studied MO theoretically using CNDO/2 method. The reaction for the abstraction from substituted-toluenes by t-butoxyl radical showed the negative ${\rho}$ values from Hammett equation, since t-butoxyl radial is electrophilic, relatively low energy SOMO, which can interact with HOMO energy of substituted-toluens. On the other hand, t-butyl radical is nucleophilic, relatively high energy SOMO, which can interact with LUMO energy of substituted-toluenes. And so the reaction of abstraction from substituted-toluenes by t-butyl radical exhibited positive ${\rho}$ values.