• Honam Mathematical Journal
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• v.35 no.3
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• pp.445-506
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• 2013

Let ${\sigma}_s(N)$ denote the sum of the s-th power of the positive divisors of N and ${\sigma}_{s,r}(N;m)={\sum_{d{\mid}N\\d{\equiv}r\;mod\;m}}\;d^s$ with $N,m,r,s,d{\in}\mathbb{Z}$, $d,s$ > 0 and $r{\geq}0$. In a celebrated paper [33], Ramanuja proved $\sum_{k=1}^{N-1}{\sigma}_1(k){\sigma}_1(N-k)=\frac{5}{12}{\sigma}_3(N)+\frac{1}{12}{\sigma}_1(N)-\frac{6}{12}N{\sigma}_1(N)$ using elementary arguments. The coefficients' relation in this identity ($\frac{5}{12}+\frac{1}{12}-\frac{6}{12}=0$) motivated us to write this article. In this article, we found the convolution sums $\sum_{k<N/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(N-mk;2)$ for odd and even divisor functions with $i,j=0,1$, $m=1,2,4$, and $d{\mid}m$. If N is an odd positive integer, $i,j=0,1$, $m=1,2,4$, $s=0,1,2$, and $d{\mid}m{\mid}2^s$, then there exist $u,a,b,c{\in}\mathbb{Z}$ satisfying $\sum_{k& lt;2^sN/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(2^sN-mk;2)=\frac{1}{u}[a{\sigma}_3(N)+bN{\sigma}_1(N)+c{\sigma}_1(N)]$ with $a+b+c=0$ and ($u,a,b,c$) = 1(Theorem 1.1). We also give an elementary problem (O) and solve special cases of them in (O) (Corollary 3.27).

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.331-360
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• 2013

Let ${\sigma}_s(N)$ denote the sum of the sth powers of the positive divisors of a positive integer N and let $\tilde{\sigma}_s(N)={\sum}_{d|N}(-1)^{d-1}d^s$ with $d$, N, and s positive integers. Hahn [12] proved that $$16\sum_{k. In this paper, we give a generalization of Hahn's result. Furthermore, we find the formula ${\sum}_{k=1}^{N-1}\tilde{\sigma}_1(2^{n-m}k)\tilde{\sigma}_3(2^nN-2^nk)$ for $m(0{\leq}m{\leq}n)$.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.1
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• pp.67-73
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• 2002

The flexibility of long reach manipulators presents a difficult control problem when accurate end-point position is required. Input shaping by convolving system commands with impulse sequences has been shown to be an effective method of reducing residual vibrations in flexible systems. However, existing shapers have been considered robustness fur only frequency uncertainty. However, this paper presents new multi-hump convolution(CV) input shaper that could accommodate with the simultaneous variation of natural frequency and damping ratio. Comparisons with previously proposed input shapers are presented to illustrate the qualities of the new input shaper. These new shapers will be shown to have better robustness fur the variation of frequency and damping ratio.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.42 no.6
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• pp.99-110
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• 2005

These days, many image processing techniques have been studied for effective image compression. Among those, The 2D image filtering is widely used for 2D image processing. The 2D image filtering can be implemented by performing the 1D linear filter separately in the horizontal and vertical direction. Efficiency of image compression depends on what filtering method is used. Generally, circular convolution is widely used in 2D image filtering for image processing. However it doesn't consider correlations at the boundary region of image, therefore effective filtering can not be performed. To solve this problem. I proposed new convolution technique using loop convolution which satisfies the 'alias-free' and 'error-free' requirement in the reconstructed image. This method could provide more effective compression performance than former methods because it used highly-correlated data when performed at the boundary region. In this paper, Sub-band Coding(SBC) was adopted to analyze efficiency of proposed filtering technique, and the simulator developed by Java-based language was used to examine the performance of proposed method.

• Honam Mathematical Journal
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• v.38 no.2
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• pp.243-257
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• 2016

If we let $L(2K;n):=\sum_{s=0}^{k-1}(^{2k}_{2s+1})\sum_{m=1}^{n-1}{\sigma}_{2k-2s-1,1}(m;2){\sigma}_{2s+1,1}(n-m;2)$ with $${\sigma}_{k,l}(n;2):=\sum\limits_{{d{\mid}n}\atop{d{\equiv}l(mod2)}}d^k$$, then we get the formula of L(2u; p)L(2v; p)L(2w; p).

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.887-902
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• 2021

In this paper, we have introduced a generalized class Sⁱ_H (m, n, 𝛾, 𝜙, 𝜓; 𝛼), i ∈ {0, 1} of harmonic univalent functions in unit disc 𝕌, a sufficient coefficient condition for the normalized harmonic function in this class is obtained. It is also shown that this coefficient condition is necessary for its subclass 𝒯 Sⁱ_H (m, n, 𝛾, 𝜙, 𝜓; 𝛼). We further obtained extreme points, bounds and a covering result for the class 𝒯 Sⁱ_H (m, n, 𝛾, 𝜙, 𝜓; 𝛼). Also, show that this class is closed under convolution and convex combination. While proving our results, certain conditions related to the coefficients of 𝜙 and 𝜓 are considered, which lead to various well-known results.

• Korean Journal of Mathematics
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• v.27 no.1
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• pp.221-267
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• 2019

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and $^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on ${\mathbb{R}}^d$. By using these operators we define the hypergeometric translation operator ${\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, and its dual $^t{\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, we express them in terms of the hypergeometric Fourier transform ${\mathcal{H}}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform ${\mathcal{H}}^W$. We study also the hypergeometric convolution product on W-invariant $L^p_{\mathcal{A}k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.12 no.11
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• pp.5555-5567
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• 2018

State-of-art instance segmentation networks are successful at generating 2D segmentation mask for region proposals with highest classification score, yet 3D object segmentation task is limited to geocentric embedding or detector of Sliding Shapes. To this end, we propose an amodal 3D instance segmentation network called A3IS-CNN, which extends the detector of Deep Sliding Shapes to amodal 3D instance segmentation by adding a new branch of 3D ConvNet called A3IS-branch. The A3IS-branch which takes 3D amodal ROI as input and 3D semantic instances as output is a fully convolution network(FCN) sharing convolutional layers with existing 3d RPN which takes 3D scene as input and 3D amodal proposals as output. For two branches share computation with each other, our 3D instance segmentation network adds only a small overhead of 0.25 fps to Deep Sliding Shapes, trading off accurate detection and point-to-point segmentation of instances. Experiments show that our 3D instance segmentation network achieves at least 10% to 50% improvement over the state-of-art network in running time, and outperforms the state-of-art 3D detectors by at least 16.1 AP.

• Proceedings of the KAIS Fall Conference
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• 2001.05a
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• pp.199-202
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• 2001

본 논문에서는 3D사운드 생성 기술 중 대표적인 방법인 원음에 HRTF(머리전달함수)를 콘볼루션(convolution)하는 방식으로 음상정위 모듈을 구현하였으며 음장감을 부여하기 위하여 잔향(reverberation) 효과를 추가하고 크로스토크 현상을 제거하기 위하여 트랜스오럴 필터를 추가하였다. 본 논문에서는 sampling rate conversion을 사용하여 decimation과 interpolation을 수행하여 44.1KHz의 sampling rate로 된 coefficient를 downsample하거나 upsample한 HRTR(머리전달함수)를 사용하여 콘볼루션(convolution)을 수행했다. 본 논문에서는 3D사운드 생성과정에서 필요한 연산과정을 최소화하여 일반 PC의 computing power로도 sampling rate conversion된 데이터를 처리하여 줄 수 있는 알고리즘을 제시하고 구현하였다.

• Smart Media Journal
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• v.13 no.6
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• pp.90-97
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• 2024

This paper focuses on detecting Alzheimer's Disease (AD). The most usual form of dementia is Alzheimer's disease, which causes permanent cause memory cell damage. Alzheimer's disease, a neurodegenerative disease, increases slowly over time. For this matter, early detection of Alzheimer's disease is important. The purpose of this work is using Magnetic Resonance Imaging (MRI) to diagnose AD. A Convolution Neural Network (CNN) model, Reset, and VGG the pre-trained learning models are used. Performing analysis and validation of layers affects the effectiveness of the model. T1-weighted MRI images are taken for preprocessing from ADNI. The Dataset images are taken from the Alzheimer's Disease Neuroimaging Initiative (ADNI). 3D MRI scans into 2D image slices shows the optimization method in the training process while achieving 96% and 94% accuracy in VGG 16 and ResNet 18 respectively. This study aims to classify AD from brain 3D MRI images and obtain better results.