• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.233-239
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• 2011

In this article we study the noton of I-monotonic sequences. We prove the decomposition theorem and generalize some of the results on monotonic sequences. We also introduce I-convergent series and studied some results.

• Journal of the Korean Statistical Society
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• v.24 no.1
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• pp.91-109
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• 1995

The rate of convergence to a random variable S for an almost certainly convergent series $S_n = \sum^n_{j=1} X_j$ of independent random variables is studied in this paper. More specifically, when $S_n$ converges to S almost certainly, the tail series $T_n = \sum^{\infty}_{j=n} X_j$ is a well-defined sequence of random variable with $T_n \to 0$ a.c. Various sets of conditions are provided so that for a given numerical sequence $0 < b_n = o(1)$, the tail series strong law of large numbers $b^{-1}_n T_n \to 0$ a.c. holds. Moreover, these results are specialized to the case of the weighted i.i.d. random varialbes. Finally, example are provided and an open problem is posed.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.481-485
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• 1996

Let G be an abelian topological group and $x_{ij} \in G$ for all $i, j \in N$ such that the series $\sum_{j=i}^{\infty} x_{ij}$ is subseries convergent for each i.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1023-1036
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• 2015

Let {Xn; $n{\succeq}1$} be a field of martingale differences taking values in a p-uniformly smooth Banach space. The paper provides conditions under which the series ${\sum}_{i{\preceq}n}\;Xi$ converges almost surely and the tail series {$Tn={\sum}_{i{\gg}n}\;X_i;n{\succeq}1$} satisfies $sup_{k{\succeq}n}{\parallel}T_k{\parallel}=\mathcal{O}p(b_n)$ and ${\frac{sup_{k{\succeq}n}{\parallel}T_k{\parallel}}{B_n}}{\rightarrow\limits^p}0$ for given fields of positive numbers {bn} and {Bn}. This result generalizes results of A. Rosalsky, J. Rosenblatt [7], [8] and S. H. Sung, A. I. Volodin [11].

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.763-772
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• 2001

The rate of convergence for an almost surely convergent series $S_n={\Sigma^n}_{i-1}X_i$ of independent random variables is studied in this paper. More specifically, when S$_{n}$ converges almost surely to a random variable S, the tail series $T_n{\equiv}$ S - S_{n-1} = {\Sigma^\infty}_{i-n} X_i$ is a well-defined sequence of random variables with T$_{n}$ $\rightarrow$ 0 almost surely. Conditions are provided so that for a given positive sequence {$b_n, n {\geq$ 1}, the limit law sup$_{\kappa}\geqn | T_{\kappa}|/b_n \rightarrow$ 0 holds. This result generalizes a result of Nam and Rosalsky [4].

• 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.

• Bulletin of the Korean Chemical Society
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• v.10 no.6
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• pp.529-535
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• 1989

A general perturbation series solution of the Smoluchowski equation is applied to investigate the rate of recombination and the remaining probability of a pair of particles in liquids. The radiative boundary condition is employed and the convergence of the perturbation series is analyzed in terms of a convergene factor in time domain. The upper bound to the error introduced by the n-th order perturbation scheme is also evaluated. The long time behaviors of the rate of recombination and the remaining probability are found to be expressed in closed forms if the perturbation series is convergent. A new and efficient method of purely numerical integration of the Smoluchowski equation is proposed and its results are compared with those obtained by the perturbation method. For the two cases where the interaction between the particles is given by (i) the Coulomb potential and (ii) the shielded Coulomb potential, the agreement between the two results is found to be excellent.

• Steel and Composite Structures
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• v.15 no.2
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• pp.175-202
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• 2013

The super convergent laminated composite beam element is newly derived for the lateral stability analysis. For this, a theoretical model of the laminated composite beams is developed based on the first-order shear deformation beam theory. The present laminated beam takes into account the transverse shear and the restrained warping induced shear deformation. The second-order coupling torque resulting from the geometric nonlinearity is rigorously derived. From the principle of minimum total potential energy, the stability equations and force-displacement relationships are derived and the explicit expressions for the displacement parameters are presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the member stiffness matrix is determined using the force-displacement relationships. In order to show accuracy and superiority of the beam element developed by this study, the critical lateral buckling moments for bisymmetric and monosymmetric I-beams are presented and compared with other results available in the literature, the isoparametric beam elements, and shell elements from ABAQUS.

• Journal of Periodontal and Implant Science
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• v.48 no.1
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• pp.22-33
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• 2018

Purpose: The purpose of this study was to evaluate the efficacy of the modified plaque score (MPS) for assessing the oral hygiene status of periodontitis patients. Methods: A total of 116 patients were included in this study. After evaluation of the $L{\ddot{o}}e$ and Silness gingival index (GI), Silness and $L{\ddot{o}}e$ plaque index (PlI), O'Leary plaque control record (PCR), and MPS, patients were randomly assigned to either a conventional tooth brushing instruction (C-TBI) group (n=56) or a professional intraoral tooth brushing instruction (P-TBI) group (n=60). The MPS and clinical parameters were re-evaluated after scaling and a series of root planing. The convergent validity of MPS with the PlI and PCR was assessed. The measurement time for MPS and PCR was compared according to the proficiency of the examiner. Results: After root planing, the GI, PlI, PCR, and MPS improved from their respective baseline values in both groups. Three different plaque indices including the MPS, showed significant differences between the C-TBI group and the P-TBI group after root planing. The MPS showed significant concurrence with the PCR and PlI. The mean time for PCR measurement was $2.76{\pm}0.71$ times longer than that for MPS measurement after 2 weeks of training. Conclusions: MPS seems to be a practical plaque scoring system compared with the PlI and PCR. These findings suggest that repetitive plaque control combined with an easily applicable plaque index (MPS) may facilitate more effective oral hygiene education and improved periodontal health.