• Korean Journal of Mathematics
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• v.13 no.2
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• pp.275-284
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• 2005

In the present paper we shall introduce several operators on the reproducing kernel spaces. And using them we shall find a solution of an infinite system of quadratic equations (1.1). In particular we shall convert problem for finding an approximate solution of infinite system of quadratic equations into problem for minimizing nonnegative biquadratic polynomial.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.471-478
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• 2007

In this article we shall introduce several operators on the reproducing kernel spaces and investigate quadratic forms on the $\mathcal{l}^2$ space. Using these operators we shall obtain a particular solution of a system of quadratic equations(1.5). Finally we can find an approximate solution of(1.5) by optimization of a nonnegative biquadratic polynomial.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.579-600
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• 2017

In this paper we consider a semilinear pseudoparabolic equation with polynomial nonlinearity and infinite delay. We first prove the existence and uniqueness of weak solutions by using the Galerkin method. Then, we prove the existence of a compact global attractor for the continuous semigroup associated to the equation. The existence and exponential stability of weak stationary solutions are also investigated.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.723-733
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• 2013

A Bachet equation $Y^2=X^3+k$ will have a rational solution if and only if there is $b{\in}\mathbb{Q}$ for which $X^3-b^2X^2+k$ is reducible. In this paper we show that such cubics arise as a cubic resolvent of a biquadratic polynomial. And we prove various properties of cubic resolvents.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.66 no.12
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• pp.1772-1781
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• 2017

In this study, the design methodology for alleviating the overfitting problem of Polynomial Neural Networks(PNN) is realized with the aid of two kinds techniques such as L2 regularization and Sum of Squared Coefficients (SSC). The PNN is widely used as a kind of mathematical modeling methods such as the identification of linear system by input/output data and the regression analysis modeling method for prediction problem. PNN is an algorithm that obtains preferred network structure by generating consecutive layers as well as nodes by using a multivariate polynomial subexpression. It has much fewer nodes and more flexible adaptability than existing neural network algorithms. However, such algorithms lead to overfitting problems due to noise sensitivity as well as excessive trainning while generation of successive network layers. To alleviate such overfitting problem and also effectively design its ensuing deep network structure, two techniques are introduced. That is we use the two techniques of both SSC(Sum of Squared Coefficients) and $L_2$ regularization for consecutive generation of each layer's nodes as well as each layer in order to construct the deep PNN structure. The technique of $L_2$ regularization is used for the minimum coefficient estimation by adding penalty term to cost function. $L_2$ regularization is a kind of representative methods of reducing the influence of noise by flattening the solution space and also lessening coefficient size. The technique for the SSC is implemented for the minimization of Sum of Squared Coefficients of polynomial instead of using the square of errors. In the sequel, the overfitting problem of the deep PNN structure is stabilized by the proposed method. This study leads to the possibility of deep network structure design as well as big data processing and also the superiority of the network performance through experiments is shown.

• Journal of the Institute of Electronics Engineers of Korea CI
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• v.48 no.5
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• pp.64-73
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• 2011

The problem of Euclidean minimum spanning tree (EMST) is to connect given nodes in a plane with minimum cost. There are many algorithms for the polynomial time problem as EMST. However, for numerous nodes, the algorithms consume an enormous amount of time to find an optimal solution. In this paper, an approximation scheme using a polynomial time approximation scheme (PTAS) algorithm with dividing and parallel processing for the problem is suggested. This scheme enables to construct a large, approximate EMST within a short duration. Although initially devised for the non-polynomial problem, we employ naive PTAS to construct a vast EMST with dynamic programming. In an experiment, the approximate EMST constructed by the proposed scheme with 15,000 input terminal nodes and 16 partition cells shows 89% and 99% saving in execution time for the serial processing and parallel processing methods, respectively. Therefore, our scheme can be applied to obtain an approximate EMST quickly for numerous input terminal nodes.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.173-195
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• 2021

A new implicit discontinuous Galerkin spectral element method (DGSEM) based on the first order hyperbolic system(FOHS) is presented for solving elliptic type partial different equations, such as the Poisson problems. By utilizing the idea of hyperbolic formulation of Nishikawa[1], the original Poisson equation was reformulated in the first-order hyperbolic system. Such hyperbolic system is solved implicitly by the collocation type DGSEM. The steady state solution in pseudo-time, which is the solution of the original Poisson problem, was obtained by the implicit solution of the global linear system. The optimal polynomial orders of 𝒪(𝒽^𝑝+1)) are obtained for both the solution and gradient variables from the test cases in 1D and 2D regular grids. Spectral accuracy of the solution and gradient variables are confirmed from all test cases of using the uniform grids in 2D.

• Journal of the Korea Institute of Military Science and Technology
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• v.18 no.1
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• pp.31-36
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• 2015

To describe turbulent wake behind a submarine, it is very important to know turbulent kinetic energy distributions in the wake. To get the distribution is to solve the turbulent kinetic energy equation, and to solve the equation, it is needed both information of ${\lambda}$ and ${\sigma}$ which define physical characteristics of the wake. This paper gives analytical solution of the equation, which is driven from $8^{th}$ order polynomial fitting, as a function of given ${\lambda}$, even though there is no information of ${\sigma}$. In comparison between numerical solution(i.e. exact solution) and analytical solution, the relative errors between them are less than to 5% in the range of 0 < ${\xi}$ < 0.95 in most given ${\lambda}$.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.20 no.2
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• pp.209-214
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• 2020

The equitable partitioning problem(EPP) is classified as [0/1] binary skill existence or nonexistence and integer skill levels such as [1,2,3,4,5]. There is well-known a polynomial-time optimal solution finding algorithm for binary skill EPP. On the other hand, tabu search a kind of metaheuristic has apply to integer skill level EPP is due to unknown polynomial-time algorithm for it and this problem is NP-hard. This paper suggests heuristic greedy algorithm with polynomial-time to find the optimal solution for integer skill level EPP. This algorithm descending sorts of skill level frequency for each field and decides the lower bound(LB) that more than the number of group, packing for each group bins first, than the students with less than LB allocates to each bin additionally. As a result of experimental data, this algorithm shows performance improvement than the result of tabu search.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.3
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• pp.243-291
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• 2020

In this paper, multi-block generalized backward differentiation methods for numerical solutions of ordinary differential and differential algebraic equations are introduced. This class of linear multi-block methods is implemented as multi-block boundary value methods (MB₂ VMs). The root distribution of the stability polynomial of the new class of methods are determined using the Wiener-Hopf factorization of a matrix polynomial for the purpose of their correct implementation. Numerical tests, showing the potential of such methods for output of multi-block of solutions of the ordinary differential equations in the new approach are also reported herein. The methods which output multi-block of solutions of the ordinary differential equations on application, are unlike the conventional linear multistep methods which output a solution at a point or the conventional boundary value methods and multi-block methods which output only a block of solutions per step. The MB₂ VMs introduced herein is a novel approach at developing very large scale integration methods (VLSIM) in the numerical solution of differential equations.