• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.397-407
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• 2006

We construct polynomial-fitting interpolation rules to agree with a function f and its first derivative f' at equally spaced nodes on the interval of interest by introducing a linear functional with which we produce systems of linear equations. We also introduce a matrix whose determinant is not zero. Such a property makes it possible to solve the linear systems and then leads to a conclusion that the rules are uniquely determined for the nodes. An example is investigated to compare the rules with Hermite interpolating polynomials.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.231-244
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• 2007

In this paper, we use measure theory for considering asymptotically stable of an autonomous system [1] of first order nonlinear ordinary differential equations(ODE's). First, we define a nonlinear infinite-horizon optimal control problem related to the ODE. Then, by a suitable change of variable, we transform the problem to a finite-horizon nonlinear optimal control problem. Then, the problem is modified into one consisting of the minimization of a linear functional over a set of positive Radon measures. The optimal measure is approximated by a finite combination of atomic measures and the problem converted to a finite-dimensional linear programming problem. The solution to this linear programming problem is used to find a piecewise-constant control, and by using the approximated control signals, we obtain the approximate trajectories and the error functional related to it. Finally the approximated trajectories and error functional is used to for considering asymptotically stable of the original problem.

• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.715-722
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• 2003

A functional central limit theorem is obtained for a stationary linear process of the form $X_{t}=\;{\Sigma_{j=0}}^{\infty}a_{j}{\epsilon}_{t-j}, where {${\in}_{t}$｝is a strictly stationary associated sequence of random variables with $E_{{\in}_t}{\;}={\;}0.{\;}E({\in}_t^2){\;}<{\;}{\infty}{\;}and{\;}{a_j}$ is a sequence of real numbers with (equation omitted). A central limit theorem for a stationary linear process generated by stationary associated processes is also discussed.

• Communications of the Korean Mathematical Society
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• v.15 no.4
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• pp.617-626
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• 2000

We study the vectors related tro states on the Cuntz algebra Ο(sub)n and prove hat, for tow states $\omega$ and $\rho$ on Ο(sub)n with $\omega$│UHF(sub)n = $\rho$│UHF(sub)n, if ($\omega$(s$_1$), …, $\omega$(s(sub)n)) and ($\rho$(s$_1$),…, $\rho$(s(sub)n)) are unit vectors, then they and linearly dependent. We also study the linear functional on Ο(sub)n associated to a sequence of unit vectors in C(sup)n which is the generalization of the Cuntz state. We show that if the linear functional associated to a sequence of unit vectors with a certain condition is a state, then it is just the Cuntz state.

• The Korean Journal of Applied Statistics
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• v.32 no.4
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• pp.607-618
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• 2019

A new and interesting task in statistics is to effectively analyze functional data that frequently comes from advances in modern science and technology in areas such as meteorology and biomedical sciences. Functional linear regression with scalar response is a popular functional data analysis technique and it is often a common problem to determine a functional association if a functional predictor variable affects the scalar response in the models. Recently, Kong et al. (Journal of Nonparametric Statistics, 28, 813-838, 2016) established classical testing methods for this based on functional principal component analysis (of the functional predictor), that is, the resulting eigenfunctions (as a basis). However, the eigenbasis functions are not generally suitable for regression purpose because they are only concerned with the variability of the functional predictor, not the functional association of interest in testing problems. Additionally, eigenfunctions are to be estimated from data so that estimation errors might be involved in the performance of testing procedures. To circumvent these issues, we propose a testing method based on fixed basis such as B-splines and show that it works well via simulations. It is also illustrated via simulated and real data examples that the proposed testing method provides more effective and intuitive results due to the localization properties of B-splines.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.643-652
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• 1995

We use a semi-variational method to obtain necessary condition for a linear functional L to attain its extrema at certain elementary products. This is applied to obtain an answer to the long-standing question of determining explicit extreme points of the normalised Spirallike functions of order $\alpha$.

• The Korean Journal of Applied Statistics
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• v.19 no.3
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• pp.431-445
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• 2006

In this paper we review some methods for analyzing functional data and illustrate real application of functional data analysis. Representing methods for functional data by using basis function, analyzing functional variation by functional principal component analysis and functional linear models are reviewed. For a real application, we use temperature and precipitation data measured in Korea from the January of 1970 to the May of 2004. We apply functional principal component analysis for each data and test the significance of regional division done by using shining hours. We also estimate functional regression model for temperature and precipitation.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.233-241
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• 2010

In this paper, we investigate the following functional inequality $${\parallel}f(\frac{x\;+\;y}{2}\;+\;z)\;+\;f(\frac{x\;+\;y}{2}\;+\;y)\;+\;f(\frac{y\;+\;z}{2}\;+\;x){\parallel\;\leq\;\parallel}2f(x\;+\;y\;+\;z)\parallel$$ in Banach modules over a $C^*$-algebra, and prove the generalized Hyers-Ulam stability of linear mappings in Banach modules over a $C^*$-algebra.

• Communications for Statistical Applications and Methods
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• v.12 no.1
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• pp.117-123
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• 2005

A functional central limit theorem is obtained for a stationary linear process of the form $X_{t}= \sum\limits_{j=0}^\infty{a_{j}x_{t-j}}$, where {x_t} is a strictly stationary sequence of negatively associated random variables with suitable conditions and {a_j} is a sequence of real numbers with $\sum\limits_{j=0}^\infty|a_{j}|<\infty$.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.119-125
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• 2007

In this paper, we investigate the following: Let A be a semisimple Banach algebra. Suppose that there exists a linear derivation $f:A{\rightarrow}A$ such that the functional equation $<f(x),x>^2=0$ holds for all $x{\in}A$. Then we have $f=0$ on A.