• Korean Journal of Mathematics
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• v.16 no.1
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• pp.1-24
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• 2008

Let $Lu=u_{tt}+u_{xxxx}$ and E be the complete normed space spanned by the eigenfunctions of L. We reveal the existence of six nontrivial solutions of a nonlinear suspension bridge equation $Lu+bu^+=1+{\epsilon}h(x,t)$ in E when the nonlinearity crosses three eigenvalues. It is shown by the critical point theory induced from the limit relative category of the torus with three holes and finite dimensional reduction method.

• Bulletin of the Korean Chemical Society
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• v.30 no.9
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• pp.2001-2006
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• 2009

Temperature dependence of the critical micelle concentration (CMC), $x_{CMC}$, in micellization can be described by ln $x_{CMC}$ = A + BT + C lnT + D/T, which has been derived statistical-mechanically. Here A, B, C, and D are fitting parameters. The equation fits the CMC data better than conventionally used polynomial equations of temperature. Moreover, it yields the unique(exponent) value of 2 when the CMC is expressed in a power-law form. This finding is quite significant, because it may point to the universality of the thermal behavior of CMC. Hence, in this article, the nature of the equation ln $x_{CMC}$ = A + BT + C lnT + D/T is examined from a lattice-theory point of view through the Flory-Huggins model. It is found that a linear behavior of heat capacity change of micellization is responsible for the CMC equation of temperature.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.471-478
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• 2024

The aim of this paper is to investigate some properties of the critical points equations on the statistical manifolds. We obtain some geometric equations on the statistical manifolds which admit critical point equations. We give a relation only between potential function and difference tensor for a CPE metric on the statistical manifolds to be Einstein.

• Korean Journal of Materials Research
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• v.25 no.8
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• pp.365-369
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• 2015

This paper investigates the dependency of the critical content for electrical conductivity of carbon powder-filled polymer matrix composites with different matrixes as a function of the carbon powder content (volume fraction) to find the break point of the relationships between the carbon powder content and the electrical conductivity. The electrical conductivity jumps by as much as ten orders of magnitude at the break point. The critical carbon powder content corresponding to the break point in electrical conductivity varies according to the matrix species and tends to increase with an increase in the surface tension of the matrix. In order to explain the dependency of the critical carbon content on the matrix species, a simple equation (${V_c}^*=[1+ 3({{\gamma}_c}^{1/2}-{{\gamma}_m}^{1/2})^2/({\Delta}q_cR]^{-1}$) was derived under some assumptions, the most important of which was that when the interfacial excess energy introduced by particles of carbon powder into the matrix reaches a universal value (${\Delta}q_c$), the particles of carbon powder begin to coagulate so as to avoid any further increase in the energy and to form networks that facilitate electrical conduction. The equation well explains the dependency through surface tension, surface tensions between the particles of carbon powder.

• Korean Journal of Mathematics
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• v.19 no.4
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• pp.423-436
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• 2011

We obtain a theorem that shows the existence of multiple solutions for the mixed type nonlinear elliptic equation with Dirichlet boundary condition. Here the nonlinear part contain the jumping nonlinearity and the subcritical growth nonlinearity. We first show the existence of a positive solution and next find the second nontrivial solution by applying the variational method and the mountain pass method in the critical point theory. By investigating that the functional I satisfies the mountain pass geometry we show the existence of at least two nontrivial solutions for the equation.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.407-414
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• 2020

In this article, a second-order Sturm-Liouville problem with impulsive effects and involving the one-dimensional p-Laplacian is considered. The existence of at least three weak solutions via variational methods and critical point theory is obtained.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.419-428
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• 2009

We show the existence of at least one nontrivial solution of the homogeneous mixed type nonlinear elliptic problem. Here mixed type nonlinearity means that the nonlinear part contain the jumping nonlinearity and the critical growth nonlinearity. We first investigate the sub-level sets of the corresponding functional in the Soboles space and the linking inequalities of the functional on the sub-level sets. We next investigate that the functional I satisfies the mountain pass geometry in the critical point theory. We obtain the result by the mountain pass method, the critical point theory and variational method.

• Korean Journal of Mathematics
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• v.20 no.2
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• pp.239-245
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• 2012

We prove that an asymmetric beam equation has at least two solutions, one of which is a positive solution. To prove the existence of the other solution, we use linking inequalities.

• Structural Engineering and Mechanics
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• v.3 no.4
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• pp.313-324
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• 1995

Flexible cantilever pipes conveying fluids with high velocity are analysed for their dynamic response and stability behaviour. The Young's modulus and mass per unit length of the pipe material have a stochastic distribution. The stochastic fields, that model the fluctuations of Young's modulus and mass density are characterized through their respective means, variances and autocorrelation functions or their equivalent power spectral density functions. The stochastic non self-adjoint partial differential equation is solved for the moments of characteristic values, by treating the point fluctuations to be stochastic perturbations. The second-order statistics of vibration frequencies and mode shapes are obtained. The critical flow velocity is first evaluated using the averaged eigenvalue equation. Through the eigenvalue equation, the statistics of vibration frequencies are transformed to yield critical flow velocity statistics. Expressions for the bounds of eigenvalues are obtained, which in turn yield the corresponding bounds for critical flow velocities.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.53-66
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• 2009

We show the existence of a nontrivial periodic solution for the superquadratic parabolic equation with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear term at infinity which have continuous derivatives. We use the critical point theory on the real Hilbert space $L_2({\Omega}{\times}(0 2{\pi}))$. We also use the variational linking theorem which is a generalization of the mountain pass theorem.