• Journal of the Korean Mathematical Society
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• v.24 no.2
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• pp.121-132
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• 1987

• Kyungpook Mathematical Journal
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• v.26 no.2
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• pp.137-148
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• 1986

• Journal of the Korean Mathematical Society
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• v.28 no.2
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• pp.275-283
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• 1991

• Proceedings of the Society of Korea Industrial and System Engineering Conference
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• 2002.05a
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• pp.371-377
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• 2002

In this paper, We consider some generalization of these five basic indices to cover non-normal distribution. The proposed generalizations are compared with the five basic indices. The results show that the proposed generalizations are more accurate than those basic indices and other generalization in measuring process capability. We compared an estimation methods by Clements with based on sample percentiles, WVM to calculate the proposed generalization as an example. The results indicated that Clements method is more accurate than percentile method, WVM in measuring process capability. But the calculations of percentile method are easy to understand, straightforward to apply, and show be valuable used for applications.

• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.569-575
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• 2006

We aim mainly at presenting two generalizations of the well-known Gauss's second summation theorem and Bailey's formula for the series $_2F_1(1/2)$. An interesting transformation formula for $_pF_q$ is obtained by combining our two main results. Relevant connections of some special cases of our main results with those given here or elsewhere are also pointed out.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.503-514
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• 2012

The concepts of t-extending and t-Baer for modules are generalized to those of FI-t-extending and FI-t-Baer respectively. These are also generalizations of FI-extending and nonsingular quasi-Baer properties respectively and they are inherited by direct summands. We shall establish a close connection between the properties of FI-t-extending and FI-t-Baer, and give a characterization of FI-t-extending modules relative to an annihilator condition.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1549-1557
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• 2015

Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially dierent ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of c-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring R, we prove that, (1) direct product of simple R-modules is c-injective; (2) an R-module D is c-injective if and only if it is isomorphic to a direct summand of a direct product of simple R-modules and injective R-modules.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.831-845
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• 2014

In this paper, we obtain new generating functions involving families of pairs of inverse functions by using a generalization of the Srivastava's theorem [H. M. Srivastava, Some generalizations of Carlitz's theorem, Pacific J. Math. 85 (1979), 471-477] obtained by Tremblay and Fug$\grave{e}$ere [Generating functions related to pairs of inverse functions, Transform methods and special functions, Varna '96, Bulgarian Acad. Sci., Sofia (1998), 484-495]. Special cases are given. These can be seen as generalizations of the generalized Bernoulli polynomials and the generalized degenerate Bernoulli polynomials.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.565-585
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• 1996

In 1976, Caristi [1] established a celebrated fixed point theorem in complete metric spaces, which is a very useful tool in the theory of nonlinear analysis. Since then, several generalizations of the theorem were given by a number of authors: for instances, generalizations for single-valued mappings were given by Downing and Kirk [4], Park [11] and Siegel [13], and the multi-valued versions of the theorem were obtained by Chang and Luo [3], and Mizoguchi and Takahashi [10].

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.449-454
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• 2007

InIn this paper we consider generalizations of the classical Dixon's theorem and the classical Whipple's theorem on the sum of a $_3F_2$. The results are derived with the help of generalized Watson's theorem obtained earlier by Mitra. A large number of results contiguous to Dixon's and Whipple's theorems obtained earlier by Lavoie, Grondin and Rathie, and Lavoie, Grondin, Rathie and Arora follow special cases of our main findings.