• 대한수학회보
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• 제55권6호
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• pp.1909-1920
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• 2018

In the paper, the authors find a common solution to three series of differential equations related to the generating function of the Bernoulli numbers of the second kind and present a recurrence relation, an explicit formula in terms of the Stirling numbers of the first kind, and a determinantal expression for the Bernoulli numbers of the second kind.

• 대한수학회보
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• 제57권5호
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• pp.1231-1240
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• 2020

The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi-polynomial behavior of graded Betti numbers of powers of homogenous ideals to ℤ-graded algebra over Noetherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.

• 대한수학회논문집
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• 제13권1호
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• pp.151-158
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• 1998

In this note we derive inequalities of linearly positive quadrant dependent random variables and obtain a strong law of large numbers for linealy positive quardant dependent random variables. Our results imply an extension of Birkel's strong law of large numbers for associated random variables to the linear positive quadrant dependence case.

• 대한수학회보
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• 제35권4호
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• pp.709-718
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• 1998

In this paper we obtain strong laws of large numbers for 2-dimensional arrays of random variables which are either pairwise positive quadrant dependent or associated. Our results imply extensions of Etemadi`s strong laws of large numbers for nonnegative random variables to the 2-dimensional case.

• 대한수학회보
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• 제43권3호
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• pp.611-617
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• 2006

In this paper we construct q-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the q-analogue of alternating sums of powers of consecutive integers due to Euler.

• 대한수학회보
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• 제38권3호
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• pp.605-610
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• 2001

It is shown that for each 1 < p < 2 there exist identically distributed uncorrelated random variables $X_n\; with\;E({$\mid$X_1$\mid$}^p)\;<\;{\infty}$, not fulfilling the weak law of large numbers (WLLN). If, however, the random variables are moreover non-negative, the weaker integrability condition $E(X_1\;log\;X_1)\;<\;{\infty}$ already guarantees the strong law of large numbers.

• 대한수학회논문집
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• 제13권2호
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• pp.385-391
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• 1998

In this note, we study a weak law of large numbers for sequences of exchangeable random variables. As a special case, we have an extension of Kolmogorov's generalization of Khintchine's weak law of large numbers to i.i.d. random variables.

• Journal of the Korean Data and Information Science Society
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• 제16권4호
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• pp.1141-1145
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• 2005

In this note, we show that Theorem 2.1[Kybernetika, 28(1992) 45-49], a result of a functional relationship between the membership function of LR fuzzy numbers of T-sum and T-product, remains valid for convex additive generator and concave shape functions L and R with simple proof. We also consider the case for 0-symmetric R fuzzy numbers.

• 대한수학회논문집
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• 제13권4호
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• pp.855-863
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• 1998

Some conditions on the strong law of large numbers for weighted sums of negative quadrant dependent random variables are studied. The almost sure convergence of weighted sums of negatively associated random variables is also established, and then it is utilized to obtain strong laws of large numbers for weighted averages of negatively associated random variables.

• 대한수학회보
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• 제56권2호
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• pp.491-499
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• 2019

The problem of finding fast computing methods for Bernoulli numbers has a long and interesting history. In this paper, the author provides new proofs for two lacunary recurrence relations with gaps of length four and eight for the Bernoulli numbers. These proofs invoked the fact that the nth powers of ${\pi}^2$, ${\pi}^4$ and ${\pi}^8$ can be expressed in terms of the nth elementary symmetric functions.