• Nonlinear Functional Analysis and Applications
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• 제28권3호
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• pp.731-751
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• 2023

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into L^r-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• 대한원격탐사학회지
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• 제33권5_1호
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• pp.455-467
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• 2017

초분광 영상을 이용한 표적 탐지를 수행할 때에는 인접한 분광 밴드의 중복성의 문제 및 고차원 데이터로 인해 발생하는 방대한 계산량의 문제점을 해결하기 위한 특징 추출 과정이 필수적이다. 본 연구는 기계 학습 분야의 특징 선택 기법을 초분광 밴드 선택에 적용하기 위해 $L_{2,1}$-norm regression 모델을 이용한 새로운 밴드 선택 기법을 제안하였으며, 제안한 밴드 선택 기법의 성능 분석을 위해 표적이 존재하는 초분광영상을 직접 촬영하고 이를 바탕으로 표적 탐지를 수행한 결과를 분석하였다. 350 nm~2500 nm 파장 대역에서 밴드 수를 164개에서 약 30~40개로 감소시켰을 때 Adaptive Cosine Estimator(ACE) 탐지 성능이 유지되거나 향상되는 결과를 보였다. 실험 결과를 통해 제안한 밴드 선택 기법이 초분광 영상에서 탐지에 효율적인 밴드를 추출해 내며, 이를 통해 성능의 감소 없이 데이터의 차원 감소를 수행할 수 있어 향후 실시간 표적 탐지 시스템의 처리 속도 향상에 도움을 줄 수 있을 것으로 보인다.

• 대한수학회보
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• 제51권1호
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• pp.139-156
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• 2014

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

• 스마트미디어저널
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• 제5권1호
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• pp.88-94
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• 2016

본 논문에서는 희소성 표현을 기반으로 하는 객체 추적 방법에 있어서 객체 표류 현상을 처리하기 위한 새로운 방법을 제시한다. 그중에서도 APG-L1 (accelerated proximal gradient L1) 방법은 희소성 표현이란 객체의 외형을 표현하기 위한 목표 템플릿(target template)과 배경이나 폐색(occlusion)과 같은 객체 이외의 부분을 대체하기 위한 기본 템플릿(trivial template)를 이용하여 입력 영상을 표현하는 방법이다. 또한 어파인 변환행렬을 이용한 particle filtering 이 적용되어 객체의 위치를 찾고 APG 방법을 사용하여 희소성기반의 L1-norm을 최소화한다. 본 논문에서는 객체추적의 표류현상을 방지하기 위하여 기본 템플릿의 계수를 활용하여 배경을 가진 객체가 채택되는 현상을 방지하는 방법을 제시한다. 다양한 영상에 적용하여 제안하는 방법을 실험한 결과, 기존의 방법들과 비교하여 높은 성과를 보인다.

• Journal of applied mathematics & informatics
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• 제32권5_6호
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• pp.707-719
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• 2014

In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in $L^2$-norm and superconvergence between the numerical solution and elliptic projection of the exact solution in $H^1$-norm or the gradient of the exact solution and recovery gradient in $L^2$-norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.

• 대한수학회지
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• 제58권1호
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• pp.69-90
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• 2021

In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ △γ(ℝ+) and Ω ∈ ����β(Sn-1) for some γ > 1 and β > 1. Here Ω ∈ ����β(Sn-1) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제15권8호
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• pp.3086-3101
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• 2021

To supply precise marketing and differentiated service for the electric power service department, it is very important to predict the customers with high sensitivity of electric power failure. To solve this problem, we propose a novel grouped 𝑙_1/2 sparsity constrained logistic regression method for sensitivity assessment of electric power failure. Different from the 𝑙₁ norm and k-support norm, the proposed grouped 𝑙_1/2 sparsity constrained logistic regression method simultaneously imposes the inter-class information and tighter approximation to the nonconvex 𝑙₀ sparsity to exploit multiple correlated attributions for prediction. Firstly, the attributes or factors for predicting the customer sensitivity of power failure are selected from customer sheets, such as customer information, electric consuming information, electrical bill, 95598 work sheet, power failure events, etc. Secondly, all these samples with attributes are clustered into several categories, and samples in the same category are assumed to be sharing similar properties. Then, 𝑙_1/2 norm constrained logistic regression model is built to predict the customer's sensitivity of power failure. Alternating direction of multipliers (ADMM) algorithm is finally employed to solve the problem by splitting it into several sub-problems effectively. Experimental results on power electrical dataset with about one million customer data from a province validate that the proposed method has a good prediction accuracy.

• 대한수학회보
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• 제32권2호
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• pp.205-209
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• 1995

Throughout this paper X is a Banach space and $\mu$ is the Lebesgue measure on [0, 1] and all operators are assumed to be bounded and linear. $L^1(\mu)$ is the Banach space of all (classes of) Lebesgue integrable functions on [0, 1] with its usual norm. Let $T : L^1(\mu) \to X$ be an operator.

• 대한수학회보
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• 제55권4호
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• pp.1209-1219
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• 2018

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky [7]: let H be the Hilbert transform and let a, b be real constants. Then for 1 < p < ${\infty}$ the norm of the operator aI + bH from $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$ is equal to $$\({\max_{x{\in}{\mathbb{R}}}}{\frac{{\mid}ax-b+(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}ax-b-(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p}{{\mid}x+{\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}x-{\tan}{\frac{\pi}{2p}}{\mid}^p}}\)^{\frac{1}{p}}$$. Our proof avoids passing through the analogous result for the conjugate function on the circle, as in [7], and is given directly on the line. We also provide new approximate extremals for aI + bH in the case p > 2.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.781-791
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• 2016

Let $d_*(1,w)^2 ={\mathbb{R}}^2$ with the octagonal norm of weight w. It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on $d_*(1,w)^2$. We also show that the unit sphere of the space of symmetric bilinear forms on $d_*(1,w)^2$ is the disjoint union of the sets of smooth points, extreme points and the set A as follows: $$S_{{\mathcal{L}}_s(^2d_*(1,w)^2)}=smB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}extB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}A$$, where the set A consists of $ax_1x_2+by_1y_2+c(x_1y_2+x_2y_1)$ with (a = b = 0, $c={\pm}{\frac{1}{1+w^2}}$), ($a{\neq}b$, $ab{\geq}0$, c = 0), (a = b, 0 < ac, 0 < ${\mid}c{\mid}$ < ${\mid}a{\mid}$), ($a{\neq}{\mid}c{\mid}$, a = -b, 0 < ac, 0 < ${\mid}c{\mid}$), ($a={\frac{1-w}{1+w}}$, b = 0, $c={\frac{1}{1+w}}$), ($a={\frac{1+w+w(w^2-3)c}{1+w^2}}$, $b={\frac{w-1+(1-3w^2)c}{w(1+w^2)}}$, ${\frac{1}{2+2w}}$ < c < ${\frac{1}{(1+w)^2(1-w)}}$, $c{\neq}{\frac{1}{1+2w-w^2}}$), ($a={\frac{1+w(1+w)c}{1+w}}$, $b={\frac{-1+(1+w)c}{w(1+w)}}$, 0 < c < $\frac{1}{2+2w}$) or ($a={\frac{1=w(1+w)c}{1+w}}$, $b={\frac{1-(1+w)c}{1+w}}$, $\frac{1}{1+w}$ < c < $\frac{1}{(1+w)^2(1-w)}$).