• Kyungpook Mathematical Journal
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• 제55권3호
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• pp.615-624
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• 2015

We present some estimates for the polynomial numerical index of $l_p$ (1 < p < ${\infty}$).

• 대한수학회지
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• 제46권6호
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• pp.1293-1307
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• 2009

We give conditions so that a polynomial be factorable through an $L_{\gamma}({\mu})$ space. Among them, we prove that, given a Banach space X and an index m, every absolutely summing operator on X is 1-factorable if and only if every 1-dominated m-homogeneous polynomial on X is right 1-factorable, if and only if every 1-dominated m-homogeneous polynomial on X is left 1-factorable. As a consequence, if X has local unconditional structure, then every 1-dominated homogeneous polynomial on X is right and left 1-factorable.

• Journal of Electrical Engineering and Technology
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• 제13권4호
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• pp.1724-1731
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• 2018

The conventional polynomial neural network (PNN) is a classical flexible neural structure and self-organizing network, however it is not free from the limitation of overfitting problem. In this study, we propose a space search-optimized polynomial neural network (ssPNN) structure to alleviate this problem. Ranking selection is realized by means of ranking selection-based performance index (RS_PI) which is combined with conventional performance index (PI) and coefficients based performance index (CPI) (viz. the sum of squared coefficient). Unlike the conventional PNN, L2-norm regularization method for estimating the polynomial coefficients is also used when designing the ssPNN. Furthermore, space search optimization (SSO) is exploited here to optimize the parameters of ssPNN (viz. the number of input variables, which variables will be selected as input variables, and the type of polynomial). Experimental results show that the proposed ranking selection-based polynomial neural network gives rise to better performance in comparison with the neuron fuzzy models reported in the literatures.

• Kyungpook Mathematical Journal
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• 제53권1호
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• pp.117-124
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• 2013

We show that for 1 < $p$ < ${\infty}$, $k$, $m{\in}\mathbb{N}$, $n^{(k)}(l_p)=inf\{n^{(k)}(l^m_p):m{\in}\mathbb{N}\}$ and that for any positive measure ${\mu}$, $n^{(k)}(L_p({\mu})){\geq}n^{(k)}(l_p)$. We also prove that for every $Q{\in}P(^kl_p:l_p)$ (1 < $p$ < ${\infty}$), if $v(Q)=0$, then ${\parallel}Q{\parallel}=0$.

• 대한원격탐사학회지
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• 제19권5호
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• pp.363-370
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• 2003

This paper describes a rectification procedure that relies on a polynomial model derived from the imaging geometry without loss of accuracy. By using polynomial model, one can effectively eliminate the iterative process to find an image pixel corresponding to each output grid point. With the imaging geometry and ephemeris data, a geo-location polynomial can be constructed from grid points that are produced by solving three equations simultaneously. And, in order to correct the local distortions induced by the geometry and terrain height, a distortion model has been incorporated in the procedure, which is a function of incidence angle and height at each pixel position. With this function, it is straightforward to calculate the pixel displacement due to distortions and then pixels are assigned to the output grid by re-sampling the displaced pixels. Most of the necessary information for the construction of polynomial model is available in the leader file and some can be derived from others. For validation, sample images of ERS-l PRI and Radarsat-l SGF have been processed by the proposed method and evaluated against ground truth acquired from 1:25,000 topography maps.

• 대한수학회지
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• 제46권5호
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• pp.919-947
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• 2009

In this paper, we give a recursive formula for the Jones polynomial of a 2-bridge knot or link with Conway normal form C($-2n_1$, $2n_2$, $-2n_3$, ..., $(-1)_r2n_r$) in terms of $n_1$, $n_2$, ..., $n_r$. As applications, we also give a recursive formula for the Jones polynomial of a 3-periodic link $L^{(3)}$ with rational quotient L = C(2, $n_1$, -2, $n_2$, ..., $n_r$, $(-1)^r2$) for any nonzero integers $n_1$, $n_2$, ..., $n_r$ and give a formula for the span of the Jones polynomial of $L^{(3)}$ in terms of $n_1$, $n_2$, ..., $n_r$ with $n_i{\neq}{\pm}1$ for all i=1, 2, ..., r.

• 대한수학회논문집
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• 제36권4호
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• pp.729-741
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• 2021

We mainly study the value distributions of L-functions in the extended selberg class. Concerning weighted sharing, we prove an uniqueness theorem when certain differential monomial of a meromorphic function share a polynomial with certain differential monomial of an L-function which improve and generalize some recent results due to Liu, Li and Yi [11], Hao and Chen [3] and Mandal and Datta [12].

• Korean Journal of Mathematics
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• 제27권1호
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• pp.17-51
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• 2019

In the paper we establish some new results depending on the comparative growth properties of composite entire and meromorphic functions using relative $_pL^*$-order, relative $_pL^*$-lower order and differential monomials, differential polynomials generated by one of the factors.

• 한국정보과학회논문지:시스템및이론
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• 제36권1호
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• pp.21-25
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• 2009

본 논문에서 우리는 평면상에 점들이 주어지는 경우에, 조각적 이차 다항식 곡선으로 맞추는 문제를 다룬다. 곡선은 이차 다항식 선분들로 이루어지고, 하나의 선분은 두 점 사이를 연결한다. 하지만 이 곡선은 점들의 부분집합만을 지나고, 지나지 못하는 점들에 대해서는 $L^{\infty}$거리로 에러를 측정한다. 이 문제에 대해서 우리는 두 가지 최적화 문제를 생각한다. 첫째로 허용 가능한 에러의 범위가 주어지고, 곡선 선분의 개수를 줄이는 문제이고, 둘째로 선분의 개수가 주어지고, 에러를 줄이는 문제이다. 주어진 점들의 개수 n에 대해서, 우리는 첫번째 문제에 대한 $O(n^2)$ 알고리즘과 두번째 문제에 대한 $O(n^3)$ 알고리즘을 제안한다.

• 전기학회논문지
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• 제66권12호
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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.