• East Asian mathematical journal
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• v.24 no.3
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• pp.223-232
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• 2008

The purpose of this paper is to prove common fixed point theorem for multivalued mappings satisfying some conditions in intuitionistic fuzzy metric space in the sense of Alaca et al.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.531-538
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• 2005

In this paper we derive the central limit theorem for ${\sum}_{i=1}^n\;a_{ni}\xi_i$, where ${a_{ni},\;1\;{\leq}\;i\;{\leq}\;n}$ is a triangular array of nonnegative numbers such that $sup_n{\sum}_{i=1}^n\;a_{ni}^2\;<\;{\infty},\;max_{1{\leq}i{\leq}n}a_{ni}{\rightarrow}0\;as\;n\;{\rightarrow}\;{\infty}\;and\;\xi'_i\;s$ are a linearly negative quadrant dependent sequence. We also apply this result to consider a central limit theorem for a partial sum of a generalized linear process $X_n\;=\;\sum_{j=-\infty}^\infty\;a_k+_j{\xi}_j$.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.2
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• pp.237-253
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• 2020

This article studies asymptotic stability of non-auto nomous linear systems with time-dependent coefficient matrices {A(t)}_t∈ℝ. The classical theorem of Levinson has been widely used to science and engineering non-autonomous systems, but systems with defective eigenvalues could not be covered because such a family does not allow continuous diagonalization. We study systems where the family allows to have upper triangulation and to have defective eigenvalues. In addition to the wider applicability, working with upper triangular matrices in place of Jordan form matrices offers more flexibility. We interpret our and earlier works including Levinson's theorem from the perspective of invariant manifold theory.

• Journal of the Korean Mathematical Society
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• v.43 no.3
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• pp.529-538
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• 2006

In this paper we derive the central limit theorem for ${\sum}^n_{i=l}\;a_{ni}{\xi}_{i},\;where\;\{a_{ni},\;1\;{\le}\;i\;{\le}n\}$ is a triangular array of non-negative numbers such that $sup_n{\sum}^n_{i=l}\;a^2_{ni}\;<\;{\infty},\;max_{1{\le}i{\le}n\;a_{ni}{\to}\;0\;as\;n{\to}{\infty}\;and\;{\xi}'_{i}s$ are a linearly positive quadrant dependent sequence. We also apply this result to consider a central limit theorem for a partial sum of a generalized linear process of the form $X_n\;=\;{\sum}^{\infty}_{j=-{\infty}}a_{k+j}{\xi}_{j}$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1149-1159
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• 2018

A quadratic polynomial ${\Phi}_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation ${\Phi}_{a,b,c}(x,y,z)=n$ has an integer solution x, y, z for any nonnegative integer n. In this article, we show that if (a, b, c) = (2, 2, 6), (2, 3, 5) or (2, 3, 7), then ${\Phi}_{a,b,c}(x,y,z)$ is universal. These were conjectured by Sun in [8].

• Journal of Integrative Natural Science
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• v.12 no.2
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• pp.44-46
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• 2019

In this paper, we study a ratio variation of Nicomachus' theorem by computing ${\sum\limits_{k=1}^{n}}[rk]$ and ${\sum\limits_{k=1}^{n}}[rk]^3$, where r is a rational number.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.235-246
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• 2010

A Banach space operator A $\in$ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A $\in$ ($\mathcal{H}\mathcal{P}$), if every part of A is polaroid. Let $X^n\;=\;\oplus^n_{t=i}X_i$, where $X_i$ are Banach spaces, and let A denote the class of upper triangular operators A = $(A_{ij})_{1{\leq}i,j{\leq}n$, $A_{ij}\;{\in}\;B(X_j,X_i)$ and $A_{ij}$ = 0 for i > j. We prove that operators A $\in$ A such that $A_{ii}$ for all $1{\leq}i{\leq}n$, and $A^*$ have the single-valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A $\in$ A such that $A_{ii}$ $\in$ ($\mathcal{H}\mathcal{P}$) for all $1{\leq}i{\leq}n$ are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, A+R satisfies Browder's theorem for all upper triangular operators R, such that $\oplus^n_{i=1}R_{ii}$ is a Riesz operator, which commutes with A.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.467-484
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• 2022

In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.561-569
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• 1997

In this paper we shall characterize a finite triangular operator matrix with M-hyponormal operators on main diagonal. This shows in particualr that such an operator is subscalar operator. As a corollary, we get that every algebraic operator is subscalar.

• 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.