• 대한수학회논문집
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• 제36권1호
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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.

• 호남수학학술지
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• 제32권1호
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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..

• 대한수학회논문집
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• 제30권3호
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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.

• 대한수학회논문집
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• 제10권3호
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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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• 제16권4호
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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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• 제7권2호
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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 $n{\geq}0$.

• 한국수학교육학회지시리즈A:수학교육
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• 제22권3호
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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.

• 충청수학회지
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• 제23권4호
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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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• 제22권2호
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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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• 제23권2호
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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.