• 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 $p$.

• Journal of Applied and Pure Mathematics
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• v.5 no.1_2
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• pp.41-53
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• 2023

Let G = (V, E) be a (p, q) graph. Define $${\rho}=\{{\frac{2}{p}},\;{\text{{\qquad} if p is even}}\\{\frac{2}{p-1}},\;{{\text{if p is odd}}$$ and L = {±1, ±2, ±3, … , ±ρ} called the set of labels. Consider a mapping f : V ⟶ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that ${\mid}{\Delta}_{f_1}-{\Delta}_{f^c_1}{\mid}{\leq}1$, where ${\Delta}_{f_1}$ and ${\Delta}_{f^c_1}$ respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs P(n, k) like P(n, 2), P(n, 3), P(n, 4).

• Applied Biological Chemistry
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• v.47 no.1
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• pp.33-40
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• 2004

Thermodynamic properties of ubiquitin transient folding intermediate were studied by measuring folding kinetics in varying temperatures and denaturant concentrations. Through quantitative kinetic modeling, the equilibrium constant, hence folding free energy, between unfolded state and intermediate state in several different temperatures were calculated. Using these values, the thermodynamic parameters were estimated. The heat capacity change $({\Delta}C_p)$ upon formation of folding intermediate from unfolded state were estimated to be around 80% of the overall folding reaction, indicating that ubiquitin folding intermediate is highly compact. At room temperature, the changes of enthalpy and entropy upon formation of the intermediate state were observed to be positive. The positive enthalpy change suggests that the breaking up of the highly ordered solvent structure surrounding hydrophobic side-chain upon formation of intermediate state. This positive enthalpy was compensated for by the positive entropy change of whole system so that formation of transient intermediate has negative free energy.

• Journal of the Korean Chemical Society
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• v.30 no.2
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• pp.188-194
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• 1986

The rate constants for the solvolysis reactions of p-chlorobenzyl chloride in ethanol-water mixtures were determinded at 30${\circ}\;and\;40{\circ}$C up to 1,600bar. Rates of reaction were increased with increasing temperature and pressure, and decreased with increasing solvent composition of ethanol mole fraction. The plots of ln k against pressure are fitted to a second-order function in P, and values of ${\Delta}V^{\neq}\;and\;${\Delta}{\beta}^{\neq}$ are obtained. The values of ${\Delta}V^{\neq}\;and\;${\Delta}{\beta}^{\neq}$ extremum behavior at about 0.20 mole fraction of ethanol. This behavior is discussed in terms of solvent structure variation. From the relation between the plots of ln k versus the solvent parameter, q ≡ (D-1)/(2D+1), or the logarithmic molar water concentration, In $C_w$, it could be estimated that the reaction proceeds through $S_N1(2)$ mechanism.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.421-426
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• 2007

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale's channel assignment problem, which was first explored by Griggs and Yeh. For positive integer $p{\geq}q$, the ${\lambda}_{p,q}$-number of graph G, denoted ${\lambda}(G;p,q)$, is the smallest span among all integer labellings of V(G) such that vertices at distance two receive labels which differ by at least q and adjacent vertices receive labels which differ by at least p. Van den Heuvel and McGuinness have proved that ${\lambda}(G;p,q){\leq}(4q-2){\Delta}+10p+38q-24$ for any planar graph G with maximum degree ${\Delta}$. In this paper, we studied the upper bound of ${\lambda}_{p,q}$-number of some planar graphs. It is proved that ${\lambda}(G;p,q){\leq}(2q-1){\Delta}+2(2p-1)$ if G is an outerplanar graph and ${\lambda}(G;p,q){\leq}(2q-1){\Delta}+6p-4q-1$ if G is a Halin graph.

• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1131-1145
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• 2012

The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with p-Laplacian type operators $$\{({\alpha}(t,x^{\Delta}(t)))^{\Delta}-a(t){\phi}_p(x^{\sigma}(t))+f({\sigma}(t),x^{\sigma}(t))=0,\;{\Delta}-a.e.\;t{\in}I\\x^{\sigma}(0)=0,\\{\beta}_1x^{\sigma}(1)+{\beta}_2x^{\Delta}({\sigma}(1))=0,$$ where ${\beta}_1$, ${\beta}_2$ > 0, $I=[0,1]^{k^2}$, ${\alpha}({\cdot},x({\cdot}))$ is an operator of $p$-Laplacian type, $\mathbb{T}$ is a time scale. Some sufficient conditions for the existence of constant-sign solutions are obtained.

• Bulletin of the Korean Chemical Society
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• v.32 no.3
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• pp.889-893
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• 2011

The energetics and transition state (TS) structures of the reactions of six substrates, $R_1R_2P$(=O or S)Cl-type where $R_1=R_2$=Me and/or MeO, with ammonia in acetonitrile are theoretically investigated at the level of CPCM-MP2/6-31+G(d) and CPCM-MP2/6-311+G(3df,2p). The degrees of distortion of TS from the ideal trigonal bipyramidal pentacoordinate, ${\Delta}{{\delta}}_{{\neq}b}$ for a backside and ${\Delta}{{\delta}}_{{\neq}f}$ for a frontside attack, are calculated. The results of calculation suggest that the feasibility of a frontside attack for P=S is greater than that for P=O system when the two ligands, $R_1$ and $R_2$, becomes larger. The experimental and calculated results of anilinolyses of $R_1R_2P$(=O or S)Cl-type show the consistent tendencies.

• Earthquakes and Structures
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• v.8 no.1
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• pp.253-273
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• 2015

The current investigation has been conducted to examine the effect of gravity loads on the seismic responses of the doubly asymmetric, three-dimensional structures comprising walls and frames. The proposed model includes the P-${\Delta}$ effects induced by the building weight. Based on the variational approach, a 3D finite element with two nodes and six DOF per node including P-${\Delta}$ effects is formulated. Dynamic and static governing equations are derived for dynamic and buckling analyzes of buildings braced by wall-frame systems. The influences of P-${\Delta}$ effects and height of the building on tip displacements under Hachinohe earthquake record are investigated through many structural examples.

• Journal of the Korean Chemical Society
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• v.32 no.5
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• pp.464-475
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• 1988

The complex formation of p-aminoazobenzene and its derivatives with Fe(III) and Mn(II) has been studied by UV and IR spectroscopy and conductometry. The effects of solvents, donor basicity, and other factors on the formation of these complexes have been examined. The vatio of metal to ligand for the complexes formed is 1 : 1, both in the solid state and in solution. The stability constants of Fe(III)-donor and Mn(II)-donor complexes are in the range of 10$^2$∼10$^4$ and 0.1∼1, respectively. The absorptivities are ~10$^4$ and ∼10$^3$ l/mol${\cdot}$cm respectively. Thermodynamic properties such as ${\Delta}H^{\circ}$, ${\Delta}G^{\circ}$ and ${\Delta}S^{\circ}$ are calculated from their stability constants utilizing Van't Hoff equation.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.777-789
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• 2016

In this paper, we establish some new oscillation criteria for the second-order forced nonlinear functional dynamic equations with damping term $$(r(t)x^{\Delta}(t))^{\Delta}+q({\sigma}(t))x^{\Delta}(t)+p(t)f(x({\tau}(t)))=e(t)$$, and $$(r(t)x^{\Delta}(t))^{\Delta}+q(t)x^{\Delta}(t)+p(t)f(x({\sigma}(t)))=e(t)$$, on a time scale ${\mathbb{T}}$, where r(t), p(t) and q(t) are real-valued right-dense continuous (rd-continuous) functions [1] defined on ${\mathbb{T}}$ with p(t) < 0 and ${\tau}:{\mathbb{T}}{\rightarrow}{\mathbb{T}}$ is a strictly increasing differentiable function and ${\lim}_{t{\rightarrow}{\infty}}{\tau}(t)={\infty}$. No restriction is imposed on the forcing term e(t) to satisfy Kartsatos condition. Our results generalize and extend some pervious results [5, 8, 10, 11, 12] and can be applied to some oscillation problems that not discussed before. Finally, we give some examples to illustrate our main results.