• Communications for Statistical Applications and Methods
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• v.25 no.3
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• pp.275-282
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• 2018

This paper discusses the use of projections for the sums of squares in the analyses of variance for two-way nested design data. The model for this data is assumed to only have random effects. Two different sizes of experimental units are required for a given experimental situation, since nesting is assumed to occur both in the treatment structure and in the design structure. So, variance components are coming from the sources of random effects of treatment factors and error terms in different sizes of experimental units. The model for this type of experimental situation is a random effects model with more than one error terms and therefore estimation of variance components are concerned. A projection method is used for the calculation of sums of squares due to random components. Squared distances of projections instead of using the usual reductions in sums of squares that show how to use projections to estimate the variance components associated with the random components in the assumed model. Expectations of quadratic forms are obtained by the Hartley's synthesis as a means of calculation.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.273-285
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• 1999

In this paper we establish the weak law of large numbers for randomly weighted partial sums of random variables and study conditions imposed on the triangular array of random weights {$W_{nj}{\;}:{\;}1{\leq}j{\leq}n,{\;}n{\geq}1$} and on the triangular array of random variables {$X_{nj}{\;}:{\;}1{\leq}j{\leq}n,{\;}{\geq}1$} which ensure that $\sum_{j=1}^{n}{\;}W_{nj}{\mid}X_{nj}{\;}-{\;}B_{nj}{\mid}$ converges In probability to 0, where {$B_{nj}{\;}:{\;}1{\;}{\leq}{\;}j{\;}{\leq}{\;}n,{\;}n{\;}{\geq}{\;}1$} is a centering array of constants or random variables.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.681-692
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• 2022

Special functions and Geometric function theory are close related to each other due to the surprise use of hypergeometric function in the solution of the Bieberbach conjecture. The purpose of this paper is to provide a set of sufficient conditions under which the normalized four parametric Wright function has lower bounds for the ratios to its partial sums and as well as for their derivatives. The sufficient conditions are also obtained by using Alexander transform. The results of this paper are generalized and also improved the work of M. Din et al. [15]. Some examples are also discussed for the sake of better understanding of this article.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.617-642
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• 2022

For positive integers n and d with d < n, an n × n array A based on 𝒳 = {0, 1, …, nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n, d), if each non-zero element of X occurs exactly once in A, and its row-sums, column-sums and two main diagonal-sums constitute a set of 2n + 2 consecutive integers. An SAMS(n, d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order n ≡ 1, 5 (mod 6), and prove that there exists a regular SAMS(n, d) for any n ≥ 5, n ≡ 1, 5 (mod 6) and d with 2 ≤ d ≤ n - 1.

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.931-957
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• 2023

For an arbitrary integer x, an integer of the form $$T(x)={\frac{x^2+x}{2}}$$ is called a triangular number. Let α₁, ... , α_k be positive integers. A sum ${\Delta}_{{\alpha}_1,{\ldots},{\alpha}_k}(x_1,\,{\ldots},\,x_k)=\{\alpha}_1T(x_1)+\,{\cdots}\,+{\alpha}_kT(x_k)$ of triangular numbers is said to be almost universal with one exception if the Diophantine equation ${\Delta}_{{\alpha}_1,{\ldots},{\alpha}_k}(x_1,\,{\ldots},\,x_k)=n$ has an integer solution (x₁, ... , x_k) ∊ ℤ^k for any nonnegative integer n except a single one. In this article, we classify all almost universal sums of triangular numbers with one exception. Furthermore, we provide an effective criterion on almost universality with one exception of an arbitrary sum of triangular numbers, which is a generalization of "15-theorem" of Conway, Miller, and Schneeberger.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.687-703
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• 2023

In this paper, under some suitable conditions, we study the Spitzer-type law of large numbers for the maximum of partial sums of independent and identically distributed random variables in upper expectation space. Some general results on necessary and sufficient conditions of the Spitzer-type law of large numbers for the maximum of partial sums of independent and identically distributed random variables under sublinear expectations are established, which extend the corresponding ones in classic probability space to the case of sub-linear expectation space.

• East Asian mathematical journal
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• v.17 no.1
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• pp.135-146
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• 2001

Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

• East Asian mathematical journal
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• v.17 no.1
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• pp.93-101
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• 2001

We give an equivalent formulation to the $Erd\"{o}s'$ conjecture on the lower bound of the greatest element in a set having distinct subset sums. On this basis we suggest a possible approach towards $ERD\"{O}S'$ conjecture. Also we reproduce L. Moser's result by means of an analytic method.

• East Asian mathematical journal
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• v.30 no.1
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• pp.31-50
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• 2014

In this paper we establish limsup results and a generalized uniform law of the iterated logarithm (LIL) for the increments of partial sums of strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) random sequences.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.55-66
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• 2014

In this paper, we study the convolution sums involving odd divisor functions, and their relations to Weierstrass ${\wp}$-functions.