• JSTS:Journal of Semiconductor Technology and Science
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• 제2권1호
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• pp.41-48
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• 2002

This paper presents a 18-mW, 2.5-㎓ fractional-N frequency synthesizer with l-bit $4^{th}$-order interpolative delta-sigma ($\Delta{\;}$\sum$)modulator to suppress fractional spurious tones while reducing in-band phase noise. A fractional-N frequency synthesizer with a quadruple prescaler has been designed and implemented in a $0.5-\mu\textrm{m}$ 15-GHz $f_t$ BiCMOS. Synthesizing 2.1 GHzwith less than 200 Hz resolution, it exhibits an in-band phase noise of less than -85 dBc/Hz at 1 KHz offset frequency with a reference spur of -85 dBc and no fractional spurs. The synthesizer also shows phase noise of -139 dBc/Hz at an offset frequency of 1.2 MHz from a 2.1GHz center frequency.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.339-344
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• 2007

The relationships between binding number and fractional edge (vertex)-deletability or fractional k-extendability of graphs are studied. Furthermore, we show that the result about fractional vertex-deletability are best possible.

• 전기전자학회논문지
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• 제3권1호
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• pp.39-49
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• 1999

This paper describes a 1 GHz, low-phase-noise CMOS fractional-N frequency synthesizer with an integrated LC VCO. The proposed frequency synthesizer, which uses a high-order delta-sigma modulator to suppress the fractional spurious tones at all multiples of the fractional frequency resolution offset, has 64 programmable frequency channels with frequency resolution of $f_ref/64$. The measured phase noise is as low as -110 dBc/Hz at a 200 KHz offset frequency from a carrier frequency of 980 MHz. The reference sideband spurs are -73.5 dBc. The prototype is implemented in a $0.5{\mu}m$ CMOS process with triple metal layers. The active chip area is about $4mm^2$ and the prototype consumes 43 mW, including the VCO buffer power consumption, from a 3.3 V supply voltage.

• 대한수학회지
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• 제53권4호
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• pp.929-967
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• 2016

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.

• Communications for Statistical Applications and Methods
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• 제5권2호
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• pp.353-359
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• 1998

Factorial designs with two-level factors represent the smallest factorial experiments. The system of notation and confounding and fractional factorial schemes developed for the $2^N$system are found in standard textbooks of experimental designs. Just as in the $2^N$ system, the general confounding and fractional factorial schemes are possible in $3^N,5^N$, .... , and $\rho^N$ where $\rho$ is a prime number. Hence, we have the $\rho^N$ system. In this article, the author proposes a new algorithm for constructing fractional factorial plans in the $\rho^N$ system.

• 대한수학회보
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• 제60권6호
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• pp.1439-1451
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• 2023

This paper provides a constructive proof of the weak factorizations of the classical Hardy space H¹(ℝⁿ) in terms of multilinear fractional integral operator on the variable Lebesgue spaces, which the result is new even in the linear case. As a direct application, we obtain a new proof of the characterization of BMO(ℝⁿ) via the boundedness of commutators of the multilinear fractional integral operator on the variable Lebesgue spaces.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.167-178
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• 2007

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

• 대한수학회논문집
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• 제31권4호
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• pp.791-809
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• 2016

By introducing two fractional convolutions, we obtain the convolution theorems for fractional Fourier cosine and sine transforms. Applying these convolutions, we construct two Boehmian spaces and then we extend the fractional Fourier cosine and sine transforms from these Boehmian spaces into another Boehmian space with desired properties.

• 대한전자공학회논문지TC
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• 제42권7호
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• pp.35-40
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• 2005

본 논문에서는 900MHz 대역 중저속 무선 통신용 칩에 이용되는 3차 ${\Delta}{\sum}$ modulator를 사용한 Fractional-N PLL 주파수 합성기를 설계 및 제작하였다 우수한 위상노이즈 특성을 얻기 위해 노이즈 특성이 좋은LC VCO를 사용하였다. 그리고 고착시간을 줄이기 위해서 Charge Pump의 펌핑 전류를 주파수 천이 값에 따라 조절할 수 있도록 제작하였고 PFD의 참조 주파수를 3MHz까지 높였다. 또한 참조 주파수를 높이는 동시에 PLL의 최소 주파수 천이 간격을 10KHz까지 줄일 수 있도록 하기위하여 36/37 Fractional-N 분주기를 제작하였다. Fractional Spur를 줄이기 위해서 3차 ${\Delta}{\sum}$ modulator를 사용하였다. 그리고 VCO, Divider by 8 Prescaler, PFD, 및 Charge Pump는 0.25um CMOS공정으로 제작되었으며, 루프 필터는 외부 컴포넌트를 이용한 3차RC 필터로 제작되었다. 그리고 Fractional-N 분주기와 3차 ${\Delta}{\sum}$ modulator는 VHDL 코드로 작성되었으며 Xilinx Spartan2E을 사용한 FPGA 보드로 구현되었다. 측정결과 PLL의 출력 전력은 약 -11dBm이고, 위상노이즈는 100kHz offset 주파수에서 -77.75dBc/Hz이다. 최소 주파수 간격은 10kHz이고, 최대 주파수 천이는 10MHz이고, 최대 주파수 변이 조건에서 고착시간은 약 800us이다.

• 대한수학회지
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• 제57권6호
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• pp.1347-1372
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• 2020

We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝⁿ, (-∆)^s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.