• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.907-914
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• 2015

In this paper, we consider a viscoelastic plate equation with p-Laplacian $u^{{\prime}{\prime}}+{\Delta}^2u-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+{\sigma}(t){\int}_{0}^{t}g(t-s){\Delta}u(s)ds-{\Delta}u^{\prime}=0$. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of both ${\sigma}$ and g.

• The Pure and Applied Mathematics
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• v.21 no.3
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• pp.147-163
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• 2014

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions $$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$) where the nonlinearity f(t, u, v) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.

• Journal of the Korean Geotechnical Society
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• v.29 no.9
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• pp.43-54
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• 2013

This study proposed a new embankment stability control method to analyze the measurement data on the slope activities of the soft ground, using the Stability Control Index (SCI) obtained from the p-q stress paths. In order to validate this new technique, the data from triaxial compression tests (CU) and field measurement were compared. SCI is calculated from the current path of the effective stress points ($p^{\prime}=p-{\Delta}u$) using the relative position between the Total Stress Path $p_{max}$ and the point of $k_f$ line $p_f$. From this result, the point of effective stress $p^{\prime}(=p-{\Delta}u)$ will have access to the point $p_f$ of $k_f$ line when the pore water pressure occurs or the point of total stress pass $p^{\prime}_{max}$ when the pore pressure dissipates. Thus, the Stability Control Index (SCI) can evaluate quantitatively the safety of embankment from the relative position of the effective stress path.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.221-229
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• 1998

Let $k$ be a real abelian field and $k_{\infty}$ be its $\mathbb{Z}_p$-extension for an odd prime $p$. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $k_n$, the $nth$ layer of the $\mathbb{Z}_p$-extension. By using the main conjecture of Iwasawa theory, we have the following: If $p$ does not divide $\prod_{{{\chi}{\in}\hat{\Delta}_k},{\chi}{\neq}1}B_{1,{\chi}{\omega}^{-1}$, then $A_n$ = {0} for all $n{\geq}0$, where ${\Delta}_k=Gal(k/\mathbb{Q})$ and ${\omega}$ is the Teichm$\ddot{u}$ller character for $p$. The converse of this statement does not hold in general. However, we have the following when $k$ is of prime conductor $q$: Let $q$ be an odd prime different from $p$. and let $k$ be a real subfield of $\mathbb{Q}({\zeta}_q)$. If $p{\mid}{\prod}_{{\chi}{\in}\hat{\Delta}_{k,p},{\chi}{\neq}1}B_{1,{\chi}{\omega}}-1$, then $A_n{\neq}\{0\}$ for all $n{\geq}1$, where ${\Delta}_{k,p}$ is the $Gal(k_{(p)}/\mathbb{Q})$ and $k_{(p)}$ is the decomposition field of $k$ for $p$.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.193-200
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• 2017

Given a pair p, q of relative prime positive integers, we have uniquely determined positive integers x, y, u and v such that vx-uy = 1, p = x + y and q = u + v. Using this property, we show that$${\sum\limits_{1{\leq}i{\leq}x,1{\leq}j{\leq}v}}\;{t^{(i-1)q+(j-1)p}\;-\;{\sum\limits_{1{\leq}k{\leq}y,1{\leq}l{\leq}u}}\;t^{1+(k-1)q+(l-1)p}$$ is the Alexander polynomial ${\Delta}_{p,q}(t)$ of a torus knot t(p, q). Hence the number $N_{p,q}$ of non-zero terms of ${\Delta}_{p,q}(t)$ is equal to vx + uy = 2vx - 1. Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary 2.8); Let q be a positive integer> 1 and let k be a positive integer. Then we have $$\begin{array}{rccl}(1)&N_{kq}+1,q&=&2k(q-1)+1\\(2)&N_{kq}+q-1,q&=&2(k+1)(q-1)-1\\(3)&N_{kq}+2,q&=&{\frac{1}{2}}k(q^2-1)+q\\(4)&N_{kq}+q-2,q&=&{\frac{1}{2}}(k+1)(q^2-1)-q\end{array}$$ where we further assume q is odd in formula (3) and (4). Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type t(kq + 2, q) and t(kq + q - 2, q) in [5] agree with $N_{kq+2,q}$ and $N_{kq+q-2,q}$ respectively.

• Journal of Adhesion and Interface
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• v.10 no.2
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• pp.84-89
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• 2009

The objective of this research was to investigate the curing behavior and adhesion performance of urea-melamine-formaldehyde (UMF) resin for the four types of UMF-1, UMF-2, UMF-3, and UMF-4 which synthesized by the staged addition of melamine. Also, various network structures of these resin types were discussed based on their different curing behavior and adhesion performance. The curing behavior was evaluated by DMTA and thermal stability was checked by TGA. Adhesion performance was evaluated by dry and wet shear strengths and the pH value of each cured resin was checked to see its effect on the adhesion performance. The results indicated that the UMF-1 resin type by the addition of melamine initially with the urea and formaldehyde at the same F/(U+M) rate showed the lowest thermal stability, rigidity (${\Delta}E^{\prime}$), temperature of tan ${\delta}$ maximum ($T_{tan}\;_{\delta}$), and wet shear strength, and pH value of cured resin. In wet shear strength, however, the UMF-4 resin type appears to be slightly higher than UMF-1 resin type.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.29 no.6C
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• pp.277-285
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• 2009

A series of undrained cyclic triaxial tests were carried out to investigate the cyclic shear stress strength characteristics of sands with respect to the silt content. Silty sand was collected around the basin of Nak-Dong River and remolded in laboratory with the range of silt content 0~50% in sand located. As results, with the change of silt content cyclic shear stress ratio (CSR) at N=10 showed the maximum value at 5% and the minimum at 20% in all relative density. The development tendency of the pore water pressure analyzed by the relationship cyclic ratio and pore water pressure ratio is unrelated the change of CSR varying silt content. Comparing the results of the void ratio and skeleton void ratio after consolidation, CSR varying silt content was much affected by skeleton void ratio which is known to affect shear behavior of silty sand.