• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.63-101
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• 2021

In this research paper, an attempt has been made to provide an interesting extension of the well-known and useful Kummer's second theorem. Several applications have also been given.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.61-71
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• 2010

The aim of this paper is to extend a number of transformation formulas for the four $X_4$, $X_5$, $X_7$, and $X_8$ among twenty triple hypergeometric series $X_1$ to $X_{20}$ introduced earlier by Exton. The results are derived from the generalized Kummer's theorem and Dixon's theorem obtained earlier by Lavoie et al..

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.227-237
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• 2015

By employing certain extended classical summation theorems, several surprising ${\pi}$ and other formulae are displayed.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.707-714
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• 1995

In this note, we extend the concepts of linearly positive quadrant dependence to the random vectors and prove a functional central limit theorem for positively quadrant dependent sequence of $R^d$-valued or separable Hilbert space valued random elements which satisfy a covariance summability condition. This result is an extension of a functional central limit theorem for weakly associated random vectors of Burton et al. to positive quadrant dependence case.

• Communications for Statistical Applications and Methods
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• v.16 no.4
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• pp.687-696
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• 2009

We prove a central limit theorem for the negatively associated random variables in a Hilbert space and extend this result to the linear process generated by negatively associated random variables in a Hilbert space. Our result implies an extension of the central limit theorem for the linear process in a real space under negative association to a simplest case of infinite dimensional Hilbert space.

• 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 $n{\geq}0$.

• The Mathematical Education
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• v.22 no.3
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• pp.27-31
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• 1984

In this paper we summarize the characteristic properties of the nuclear space, and then try to establish the relation between Hopf's extension theorem and nuclear space on $\sigma$-Hilbert space.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.645-656
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• 2010

In this paper, we prove a common fixed point theorem for four mappings on fuzzy 2-metric spaces. Our result is an extension of results of S. H. Cho [2] to fuzzy 2-metric spaces. Also, it is a generalization of a result of S. Sharma [11].

• Journal of the Korean Statistical Society
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• v.22 no.2
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• pp.353-359
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• 1993

In this note we consider other type of tightness than that of Birkel (1988) and prove an invariance principle for nonstationary associated processes by an application of the central limit theorem of Cox and Grimmett (1984), thus avoiding the argument of uniform integrability. This result is an extension to the nonstationary case of an invariance priciple of Newman and Wright (1981) as well as an improvement of the central limit theorem of Cox and Grimmett (1984).

• Journal of the Korean Statistical Society
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• v.23 no.2
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• pp.504-512
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• 1994

A simple proof for the random central limit theorem is given for a family of stationary linear lattice processes, which belogn to a class of 2 dimensional random fields, applying the Beveridge and Nelson decomposition in time series context. The result is an extension of Fakhre-Zakeri and Fershidi (1993) dealing with the linear process in time series to the case of the linear lattice process with 2 dimensional indices.