• 대한전자공학회논문지SD
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• 제43권10호
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• pp.90-96
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• 2006

본 논문에서는 capacitance scaling 구조를 이용하여 짧은 locking 시간과 작은 fractional spur를 가지는 ${\Sigma}{\Delta}$ fractional-N PLL을 설계하였다. 루프필터의 실효 커패시턴스를 변화시키기 위하여 여러 개의 전하펌프를 이용해 서로 다른 경로로 커패시터에 전류를 공급하였다. 필터의 실효 커패시턴스는 동작상태에 따라 크기가 변하며 커패시터들은 하나의 PLL 칩에 집적화 할 수 있을 정도로 작은 크기를 가진다. 또한 PLL이 lock 되면 전하펌프 전류의 크기도 작아져 fractional spur의 크기도 작아진다. 제안된 구조는 HSPICE CMOS $0.35{\mu}m$ 공정으로 시뮬레이션 하였으며 $8{\mu}s$ 이하의 locking 시간을 가진다. PLL의 루프필터는 200pF, 17pF의 작은 커패시터와 $2.8k{\Omega}$의 저항으로 설계되었다.

• 대한수학회논문집
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• 제11권1호
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• pp.139-145
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• 1996

We show how some interesting results involving series summation and the digamma function are established by means of Riemann-Liouville operator of fractional calculus. We derive the relation $$ \frac{\Gamma(\lambda)}{\Gamma(\nu)} \sum^{\infty}_{n=1}{\frac{\Gamma(\nu+n)}{n\Gamma(\lambda+n)}_{p+2}F_{p+1}(a_1, \cdots, a_{p+1},\lambda + n; x/a)} = \sum^{\infty}_{k=0}{\frac{(a_1)_k \cdots (a_{(p+1)}{(b_1)_k \cdots (b_p)_k K!} (\frac{x}{a})^k [\psi(\lambda + k) - \psi(\lambda - \nu + k)]}, Re(\lambda) > Re(\nu) \geq 0 $$ and explain some special cases.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.695-703
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• 2002

The fractional order evolutionary integral equations have been considered by first author in [6], the existence, uniqueness and some other properties of the solution have been proved. Here we study the continuation of the solution and its fractional order derivative. Also we study the generality of this problem and prove that the fractional order diffusion problem, the fractional order wave problem and the initial value problem of the equation of evolution are special cases of it. The abstract diffusion-wave problem will be given also as an application.

• JSTS:Journal of Semiconductor Technology and Science
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• 제7권4호
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• pp.267-273
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• 2007

A fractional-N frequency synthesizer supports quadruple bands and multiple standards for mobile broadcasting systems. A novel linearized coarse tuned VCO adopting a pseudo-exponential capacitor bank structure is proposed to cover the wide bandwidth of 65%. The proposed technique successfully reduces the variations of KVCO and per-code frequency step by 3.2 and 2.7 times, respectively. For the divider and prescaler circuits, TSPC (true single-phase clock) logic is extensively utilized for high speed operation, low power consumption, and small silicon area. Implemented in $0.18-{\mu}m$ CMOS, the PLL covers $154{\sim}303$ MHz (VHF-III), $462{\sim}911$ MHz (UHF), and $1441{\sim}1887$ MHz (L1, L2) with two VCO's while dissipating 23 mA from 1.8 V supply. The integrated phase noise is 0.598 and 0.812 degree for the integer-N and fractional-N modes, respectively, at 750 MHz output frequency. The in-band noise at 10 kHz offset is -96 dBc/Hz for the integer-N mode and degraded only by 3 dB for the fractional-N mode.

• 대한수학회보
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• 제51권2호
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• pp.329-338
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• 2014

In this paper, we study the method of upper and lower solutions and develop the generalized quasilinearization technique for the existence and approximation of solutions to some three-point nonlocal boundary value problems associated with higher order fractional differential equations of the type $$^c{\mathcal{D}}^q_{0+}u(t)+f(t,u(t))=0,\;t{\in}(0,1)$$ $$u^{\prime}(0)={\gamma}u^{\prime}({\eta}),\;u^{\prime\prime}(0)=0,\;u^{\prime\prime\prime}(0)=0,{\ldots},u^{(n-1)}(0)=0,\;u(1)={\delta}u({\eta})$$, where, n-1 < q < n, $n({\geq}3){\in}\mathbb{N}$, 0 < ${\eta},{\gamma},{\delta}$ < 1 and $^c\mathcal{D}^q_{0+}$ is the Caputo fractional derivative of order q. The nonlinear function f is assumed to be continuous.

• 대한수학회지
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• 제38권6호
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• pp.1191-1204
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• 2001

The paper is devoted to the study of fractional integration and differentiation on a finite interval [a, b] of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration $\int_{a}^{x}(t/x)^{\mu}f(t)dt/t$ and the modified differentiation ${\delta}+{\mu}({\delta}=xD,D=d/dx)$ with real $\mu$, being taken n times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space $X^{p}_{c}$(a, b) of Lebesgue measurable functions f on $R_{+}=(0,{\infty})$ such that for c${\in}R=(-{\infty}{\infty})$, in particular in the space $L^{p}(0,{\infty})\;(1{\le}{\le}{\infty})$. The existence almost every where is established for the coorresponding Hadamard-type fractional derivative for a function g(x) such that $x^{p}$g(x) have $\delta$ derivatives up to order n-1 on [a, b] and ${\delta}^{n-1}[x^{\mu}$g(x)] is absolutely continuous on [a, b]. Semigroup and reciprocal properties for the above operators are proved.

• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.19-32
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• 2005

It is shown that the problem of minimizing (maximizing) a quadratic cost functional (quadratic gain functional) given the dynamics dx = (fx + gu)dt + hdb(t, a) where b(t, a) is a fractional Brownian motion of order a, 0 < 2a < 1, can be solved completely (and meaningfully!) by using the dynamical equations of the moments of x(t). The key is to use fractional Taylor's series to obtain a relation between differential and differential of fractional order.

• East Asian mathematical journal
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• 제14권1호
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• pp.107-116
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• 1998

This paper deals with functions of the form $f(z)=a_1z-{\sum}{\limits}_{n=2}^{\infty}a_nz^n(a_1>0,\;a_n{\geqslant}0)$ with $(1-{\mu})f(z_0)/z_0+{\mu}f'(z_0)=1(-1. We introduce the class $\varphi({\mu},{\eta},{\gamma},{\delta},A,B;\;z_0)$ with generalized fractional derivatives. Also we have obtained coefficient inequalities, distortion theorem and radious problem of functions belonging to the calss $\varphi({\mu},{\eta},{\gamma},{\delta},A,B;\;z_0)$.

• Korean Journal of Mathematics
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• 제29권2호
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• pp.211-225
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• 2021

In the present paper we introduce some sequence spaces over n-normed spaces defined by fractional difference operator and Musielak-Orlicz function 𝓜 = (𝕱_i). We also study some topological properties and prove some inclusion relations between these spaces.

• 대한수학회논문집
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• 제37권4호
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• pp.1099-1129
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• 2022

In this paper, we establish Liouville type theorems for the fractional powers of multidimensional Bessel operators extending the results given in [6]. In order to do this, we consider the distributional point of view of fractional Bessel operators studied in [12].