• The Pure and Applied Mathematics
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• v.7 no.2
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• pp.79-86
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• 2000

In this paper, we introduce the classes H(p,q,k),K(p;k) of operators determined by the Heinz-Kato-Furuta inequality and Holer-McCarthy inequality. We characterize relationship between p-quasihyponormal, $\kappa$-quasihyponormal and $\kappa$-p-quasihyponormal operators. And it is proved that every operator in K(p;1) for some $0 is paranormal.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.227-233
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• 1999

Let $k$ be a real abelian field and $k_{\infty}={\bigcup}_{n{\geq}0}k_n$ be its $\mathbb{Z}_p$-extension for an odd prime $p$. For each $n{\geq}0$, we denote the class number of $k_n$ by $h_n$. The following is a well known theorem: Theorem. Suppose $p$ remains inert in $k$ and the prime ideal of $k$ above $p$ totally ramifies in $k_{\infty}$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$ for all $n{\geq}0$. The aim of this paper is to generalize above theorem: Theorem 1. Suppose $H^1(G_n,E_n){\simeq}(\mathbb{Z}/p^n\mathbb{Z})^l$, where $l$ is the number of prime ideals of $k$ above $p$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$. Theorem 2. Let $k$ be a real quadratic field. Suppose that $H^1(G_1,E_1){\simeq}(\mathbb{Z}/p\mathbb{Z})^l$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$ for all $n{\geq}0$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1079-1095
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• 2021

In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo p². One of our results is as follows: for prime number p > 3, $${\sum\limits_{k=(p+1)/2}^{p-1}}\;k^2B_{p,k}B_{p,k-(p-1)/2}H_k{\equiv}(-1)^{(p-1)/2}\(-{\frac{521}{36}}p-{\frac{1}{p}}-{\frac{41}{12}}+pH^2_{3(p-1)/2}-10pq^2_p(2)+4\({\frac{10}{3}}p+1\)q_p(2)\)\;(mod\;p^2),$$ where q_p(2) is Fermat quotient.

• 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 $p$.

• Horticultural Science & Technology
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• v.33 no.6
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• pp.854-859
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• 2015

This experiment was performed to develop a model for nutrition ion concentration and EC in regard to change in pH from 4.0 to 8.0 in nutrient solution. The pH changes according to the variation of $HPO_4{^{-2}}$ and $H_2PO_4{^-}$ in the nutrient solution while variation of EC increased from pH 4.0 to 5.0, stabilized from pH 5.0 to 7.0 and increased again from pH 7.0 to 8.0. For the variance of major elements in the nutrient solution, K, Ca, N and P increased while pH was higher, especially the variables for K and P were increased largely. On the other hand, variables of Mg and S were stable. Based on analysis of the ion balance model of nutrient solution, the cation increased than anion over rising the variation of pH while balance point of ion moved from a-side to d-side. In addition, the imbalance increased while it moved away from the EC centerline as variance of pH increased. It was larger than effect of EC variance to correction values of equivalence ratios of K and Ca about variation of $HPO_4{^{-2}}$ and $H_2PO_4{^-}$ while as variance of pH increased, K decreased but Ca increased. These showed the result that variance of pH about correction values of equivalence ratios of K and Ca gave a second-degree polynomial model rating of 0.97. Through this research, it was identified the pH variable model about variance among pH, ion and EC according to gradient of phosphate.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.83-96
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• 2013

Let K be a number field and fix a prime number $p$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $B_{K,p}$ of primes of K satisfying that any elliptic curve over K with $B_{K,p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $B_{K,p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of E gives its associated Galois representation $\bar{\rho}_{E,p}$ modulo $p$. We also study the irreducibility and surjectivity of $\bar{\rho}_{E,p}$ for semistable elliptic curves with $B_{K,p}$-reduction.

• The Korean Journal of Mycology
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• v.20 no.4
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• pp.296-301
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• 1992

Species of genus Pluteus were cellected in Korea from 1984 to 1991. Eight species in Pluteus were identified, and six species were confirmed new to Korea and described here; Plutens atromarginatus, P. spinulosus, P. patricius, P. depauperatus, P. satur, and P. chrysophlebius ssp. sublaevigatus.

• Honam Mathematical Journal
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• v.16 no.1
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• pp.119-135
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• 1994

Let $Q_{n+p-1}(\alpha)$ denote the- dass of functions $$f(z)=z^{P}-\sum_{n=0}^\infty{a_{(p+k)}z^{p+k}$$ ($a_{p+k}{\geq}0$, $p{\in}N=\left{1,2,{\cdots}\right}$) which are analytic and p-valent in the unit disc $U=\left{z:{\mid}z:{\mid}<1\right}$ and satisfying $Re\left{\frac{D^{n+p-1}f(\approx))^{\prime}}{pz^{p-a}\right}>{\alpha},0{\leq}{\alpha}<1,n>-p,z{\in}U.$ In this paper we obtain sharp results concerning coefficient estimates, distortion theorem, closure theorems and radii of p-valent close-to- convexity, starlikeness and convexity for the class $Q_{n+p-1}$ ($\alpha$). We also obtain class preserving integral operators of the form $F(z)=\frac{c+p}{z^{c}}\int_{o}^{z}t^{c-1}f(t)dt.$ c>-p $F\left(z\right)=\frac{c+p}{z^{c}}\int_{0}^{z} t^{c-1}f\left(t \right)dt. \qquad c>-p$ for the class $Q_{n+p-1}$ ($\alpha$). Conversely when $F(z){\in}Q_{n+p-1}(\alpha)$, radius of p-valence of f(z) has been determined.

• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.271-283
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• 2018

In this paper, we introduce various first order (p, q)-difference equations. We investigate solutions to equations which are linear (p, q)-difference equations and nonlinear (p, q)-difference equations. We also find some properties of (p, q)-calculus, exponential functions, and inverse function.

• Journal of The Korean Society of Grassland and Forage Science
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• v.32 no.1
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• pp.15-28
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• 2012

This study was carried out to know adaptability and growth characteristics, yield, chemical compositions and nutrition yield of corn hybrids for silage at paddy field. The field experiments were conducted at Sangju province for one year (2009). The experimental design was arranged in a randomized block design with three replication. The treatments consisted of eleven corn hybrids. The planting date was on 1 May and harvested at 24 August. Stem diameter, stem hardness and number of ear were higher in P32P75 than other varieties. Ear height, dead leaf and green degree were highest in $NC^+$7117, but number of root system and Brix ($B^{\circ}$) were higher in P3394 than other varieties. Crude protein and crude fat (EE) were highest in P32K61 and P31P41, respectively (P<0.01). NDF and ADF were highest in KPO and KIO, respectively, but no significant differences were found among the varieties. Total mineral contents were the highest in Kwangpyongok (9,775 mg/kg), and P3394 (6,651 mg/kg) was the lowest as compared to other varieties (P<0.01). Crude protein yield, crude fat yield and mineral yield were highest in P3156, P31P41 and KPO, respectively (P<0.01). Total composition amino acid and total fatty acid were the highest in P32K61 and KIO, respectively (P<0.01). Yields of crude protein, fatty acid, composition amino acid and TDN were the highest in P3156 (P<0.01). But yields of crude protein and mineral were the highest in P31P41 and KPO, respectively (P<0.01). Total digestible nutrient (TDN) was higher in order of P3156 > $NC^+$7117 > P31N27 > KPO > P32K61 > P32T83 > P32P75 > P31P41 > P3394 > P32W86 > KIO. Based on the above results, corn hybrid varieties could be recommended in P3156, NC+7117 and P31N27 for growth characteristics, quantitative production and nutrition yield.