• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1511-1528
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• 2020

A skew generalized power series ring R[[S, 𝜔]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action 𝜔 of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal'cev-Neumann series rings, the "untwisted" versions of all of these, and generalized power series rings. In this paper we obtain some necessary conditions on R, S and 𝜔 such that the skew generalized power series ring R[[S, 𝜔]] is (uniquely) clean. As particular cases of our general results we obtain new theorems on skew Mal'cev-Neumann series rings, skew Laurent series rings, and generalized power series rings.

• Proceedings of the Korean Statistical Society Conference
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• 2002.05a
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• pp.49-54
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• 2002

For arbitrary random variables {$X_{n},n{\geq}1$}, the order of growth of the series. $S_{n}\;=\;{\sum}_{j=1}^n\;X_{j}$ is studied in this paper. More specifically, when the series S_{n}$ diverges almost surely, the strong law of large numbers $S_{n}/g_{n}^{-1}$($A_{n}{\psi}(A_{n}))\;{\rightarrow}\;0$ a.s. is constructed by extending the results of Petrov (1973). On the other hand, if the series $S_{n}$ converges almost surely to a random variable S, then the tail series $T_{n}\;=\;S\;-\;S_{n-1}\;=\;{\sum}_{j=n}^{\infty}\;X_{j}$ is a well-defined sequence of random variables and converges to 0 almost surely. For the almost surely convergent series $S_{n}$, a tail series strong law of large numbers $T_{n}/g_{n}^{-1}(B_{n}{\psi}^{\ast}(B_{n}^{-1}))\;{\rightarrow}\;0$ a.s., which generalizes the result of Klesov (1984), is also established by investigating the duality between the limiting behavior of partial sums and that of tail series. In particular, an example is provided showing that the current work can prevail despite the fact that previous tail series strong law of large numbers does not work.

• Journal of the Korean Society of Fashion and Beauty
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• v.1 no.1 s.1
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• pp.27-47
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• 2003

This research is to build the foundation of systematic application of color in cosmetology by analyzing color attributes in women's makeup presentation. The result were as follows. 1. The most popular color series in make up were R then RP and YR. The most popular color tone is 'd' and 'lt'. 2. Colors in make up according to age was as follows. For eye shadow, people aged 18 to 24 used 'lt' tone of the R color series; people aged 25 to 34 used 'lt', 's', 'sf tone of the R color series, 'lt' tone of the PB color series, 'lt' tone of the YR color series; people over 35 'g' tone of the YR color series, 'sf' tone of the P color series. For lipstick, people aged 18 to 24 used 'd' tone of the R color series; people aged 25 to 34 used 'd', 'sf' tone of the R color series; people over 35 used 'd' tone of the R color series. For lip-gloss, people aged 18 to 24 used 'v', 'lt', 'b', 's' tone of the R color series; people aged 25 to 34 used 's' 'd' 'dp' 'sf' tone of the R color series; people over 35 used 'b' tone of the R color series. 3. Make up colors according to marital status was as follows. For eye shadow, while married interviewees used 's', 'dk' tone of the R color series, single interviewees used 'lt', 'sf' tone of the R color series. For lipstick, while married interviewees used 'd', 'g' tone of the R color series, single interviewees preferred to use madder 'd', 'sf' tone of the R color series. For lip-gross, while married interviewees used 'd' tone of the R color series, single interviewees used 'b' tone of the R color series the most.

• The Journal of the Korea Contents Association
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• v.6 no.5
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• pp.29-34
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• 2006

For the almost certainly convergent series $S_n=\sum_{i=1}^nV-i$ of independent random elements in Banach spaces, by investigating tail series laws of large numbers, the rate of convergence of the series $S_n$ to a random variable s is studied in this paper. More specifically, by studying the duality between the limiting behavior of the tail series $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}V-i$ of random variables and that of Banach space valued random elements, an alternative way of proving a result of the previous work, which establishes the equivalence between the tail series weak law of large numbers and a limit law, is provided in a Banach space setting.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.479-494
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• 2016

In this paper, using a transformation formula for a certain series which comes from the generalized non-analytic Eisenstein series, we obtained some infinite series identities which contain Ramanujan's formula and author's previous results.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.363-377
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• 2015

Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

• The Journal of the Korea Contents Association
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• v.6 no.4
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• pp.63-68
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• 2006

For the almost co티am convergent series $S_n$ of independent random variables, by investigating the limiting behavior of the tail series, $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}X_i$, the rate of convergence of the series $S_n$ to a random variable S is studied in this paper. More specifically, the equivalence between the tail series weak law of large numbers and a limit law is established for a quasi-monotone decreasing sequence, thereby extending a result of Previous work to the wider class of the norming constants.

• Elastomers and Composites
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• v.29 no.1
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• pp.18-29
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• 1994

The purpose of this study is to investigate reinforced effect between silica treated by coupling agents and rubber matrix under the configuration chemical bonds, and the effect of silica particles coated by organic polymers using aluminum chloride as the catalyst. In vulcanization characteristies were tested by Curastometer. The M-series vulcanizates were reached to the fastest optimum cure $time(t_{90})$ and R-series vulcanizates with the same formula had the shorted optimum cure times. Tensile characteristics measuring with a tensile tester revealed that the M-series vulcanizate was the best in the physical properties, such as tensile strength. In 100% modulus, however, the S-series vulcanizates appeared to be better than the other vulcanizates. Also, hardness showed the following order : S-series>R-series>M-series with the order of elongation R-series>M-series>S-series. In SEM test, shapes of chemical treated silicas were observed. The dispersion of filler in the SBR composite appeard uniformly. In RDS test for the dynamic characteristics, G' indicates that S-3 shows the highest value with the next order M-3>R-3, and the order of damping values are as followe: M-3>M-3>R-3.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1201-1211
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• 2009

The aim of this research paper is to establish generalizations of classical Dixon's theorem for the series $_3F_2$, a result due to Bailey involving product of generalized hypergeometric series and certain very interesting summations due to Ramanujan. The results are derived with the help of generalized Kummer's summation theorem for the series $_2F_1$ obtained earlier by Lavoie, Grondin, and Rathie.