• Title/Summary/Keyword: Dirac delta

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SINGULAR SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS IN SEVERAL SPACE DIMENSIONS

  • Baek, Jeong-Seon;Kwak, Min-Kyu
    • Journal of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1049-1064
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    • 1997
  • We study the existence and uniqueness of nonnegative singular solution u(x,t) of the semilinear parabolic equation $$ u_t = \Delta u - a \cdot \nabla(u^q) = u^p, $$ defined in the whole space $R^N$ for t > 0, with initial data $M\delta(x)$, a Dirac mass, with M > 0. The exponents p,q are larger than 1 and the direction vector a is assumed to be constant. We here show that a unique singular solution exists for every M > 0 if and only if 1 < q < (N + 1)/(N - 1) and 1 < p < 1 + $(2q^*)$/(N + 1), where $q^* = max{q, (N + 1)/N}$. This result agrees with the earlier one for N = 1. In the proof of this result, we also show that a unique singular solution of a diffusion-convection equation without absorption, $$ u_t = \Delta u - a \cdot \nabla(u^q), $$ exists if and only if 1 < q < (N + 1)/(N - 1).

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Influence of impulsive line source and non-homogeneity on the propagation of SH-wave in an isotropic medium

  • Kakar, Rajneesh
    • Interaction and multiscale mechanics
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    • v.6 no.3
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    • pp.287-300
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    • 2013
  • In this paper, the effect of impulsive line on the propagation of shear waves in non-homogeneous elastic layer is investigated. The rigidity and density in the intermediate layer is assumed to vary quadratic as functions of depth. The dispersion equation is obtained by using the Fourier transform and Green's function technique. The study ends with the mathematical calculations for transmitted wave in the layer. These equations are in complete agreement with the classical results when the non-homogeneity parameters are neglected. Various curves are plotted to show the effects of non-homogeneities on shear waves in the intermediate layer.

Vibration Analysis of Rotating Cantilever Plates with a Concentrated Mass (집중 질량을 가진 회전하는 외팔 평판의 진동 해석)

  • 양정식;유홍희
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 1998.04a
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    • pp.181-186
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    • 1998
  • A modelling method for the vibration analysis of rotating cantilever plates with a concentrated mass is presented. The equations of motion for the rotating plates with a concentrated mass located in an arbitrary position are derived. For the modelling of the concentrated mass, a mass density Dirac delta function is used. The effects of concentrated mass and its location, angular speed, and hub radius of the rotating plate on the natural frequencies are studied. Particularly, mode shape variations due to some parameter variations are investigated.

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ON THE PARAMETIC INTEREST OF THE BLACK-SCHOLES EQUATION

  • Kananthai, Amnuay
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.923-929
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    • 2010
  • We have discovered some parametics $\lambda$ in the Black-Scholes equation which depend on the interest rate $\gamma$ and the Volatility $\sigma$ and later is named the parametic interest. On studying the parametic interest $\lambda$, we found that such $\lambda$ gives the sufficient condition for the existence of solutions of the Black-Scholes equation which is either weak or strong solutions.

Forecast Modeling of Catalyst Deactivation in Coal Liquefaction (석탄 액화반응에서의 촉매 불활성에 관한 예측 모델링)

  • 이영우;손재익
    • Journal of Energy Engineering
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    • v.3 no.1
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    • pp.18-27
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    • 1994
  • 석탄액화반응에서 촉매 세공구조가 촉매 불활성화에 미치는 영향을 조사하기 위하여 간단한 모델을 전개하였다. 촉매의 세공수 분포에 근거하여 두 개의 Dirac delta 함수분포를 갖는 다공질 촉매구조를 제안하였으며 촉매 세공구조와 반응속도상수와의 관계를 유도하기 위하여 단순화된 반응계를 가정하였다. 균일 코드피복 가정에서 본 모델을 촉매 불활성화 예측에 적용하였으며 계산과정에서 세공율, 세공 크기 등의 촉매 특성치에 대해서는 실제값을 이용하였다. 본 모델연구에 의하면 unimodal 촉매에 비해 bimodal 촉매가 촉매 불활성화에 덜 민감하였다.

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Vibration Anatysis on plates Stffened wlth Viscoelastic Beams (점탄성보로 보강된 평판의 진동해석)

  • Choi, Jang-Woo;Jung, Seok-Ju;Jung, Kang
    • Journal of the Korean Society of Safety
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    • v.4 no.1
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    • pp.47-58
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    • 1989
  • This paper treats the vibration analysis of a simply supported rectangular plate stiffened with viscoelastic beams. The effect of viscoelastic beams on the vibration of the plate is analyzed by using Dirac delta function and the equation of motion can be expressed only one equation. The frequency equation is obtained by applying Laplace transformation. The effect of volumes, numben and aspect ratios of beam on the frequency of the plate is analyzed.

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SH-wave propagation in a heterogeneous layer over an inhomogeneous isotropic elastic half-space

  • Kakar, Rajneesh
    • Earthquakes and Structures
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    • v.9 no.2
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    • pp.305-320
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    • 2015
  • The present paper is devoted to study SH-wave propagation in heterogeneous layer laying over an inhomogeneous isotropic elastic half-space. The dispersion relation for propagation of said waves is derived with Green's function method and Fourier transform. As a special case when the upper layer and lower half-space are homogeneous, our derived equation is in agreement with the general equation of Love wave. Numerically, it is observed that the velocity of SH-wave increases with the increase of inhomogeneity parameter.

Two-Dimensional Probability Functions of Morphological Dilation and Erosion of a Memoryless Source

  • Sangsin Na;Park, Tae-Young
    • Journal of Electrical Engineering and information Science
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    • v.1 no.1
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    • pp.151-155
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    • 1996
  • This paper derives the two-dimensional probability distribution and density functions of morphological dilation and erosion of a one-dimensional memoryless source and reports numerical results for a uniform source, thus providing methodology for joint distributions for other morphological operations. The joint density functions expressed in closed forms contain the Dirac delta functions due to the joint discontinuity within the dilation and erosion. They also exhibit symmetry between these two morphological density functions of dilated and/or eroded sources, in the computation of other higher moments thereof, and in multidimensional quantization.

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A New Bussgang Blind Equalization Algorithm with Reduced Computational Complexity (계산 복잡도가 줄어든 새로운 Bussgang 자력 등화 알고리듬)

  • Kim, Seong-Min;Kim, Whan-Woo
    • The Journal of Korean Institute of Electromagnetic Engineering and Science
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    • v.22 no.10
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    • pp.1012-1015
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    • 2011
  • The decision-directed blind equalization algorithm is often used due to its simplicity and good convergence property when the eye pattern is open. However, in a channel where the eye pattern is closed, the decision-directed algorithm is not guaranteed to converge. Hence, a modified Bussgang-type algorithm using a hyperbolic tangent function for zero-memory nonlinear(ZNL) function has been proposed and applied to avoid this problem by Filho et al. But application of this algorithm includes the calculation of hyperbolic tangent function and its derivative or a look-up table which may need a large amount of memory due to channel variations. To reduce the computational and/or hardware complexity of Filho's algorithm, in this paper, an improved method for the decision-directed algorithm is proposed. In the proposed scheme, the ZNL function and its derivative are respectively set to be the original signum function and a narrow rectangular pulse which is an approximation of Dirac delta function. It is shown that the proposed scheme, when it is combined with decision-directed algorithm, reduces the computational complexity drastically while it retains the convergence and steady-state performance of the Filho's algorithm.

Derivation of the Transient. Solution of the Horizontal Interfacial Electric Field Generated by a Tiny Horizontal Current Source on a Uniaxially Anisotropic Half-Space Dielectric (단축 이방성 반공간 유전체 표면에서 수평 미세 전류원에 의해 발생하는 수평 경계면 전기장에 대한 시영역 해의 유도)

  • 이원석;남상욱
    • The Journal of Korean Institute of Electromagnetic Engineering and Science
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    • v.11 no.2
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    • pp.313-321
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    • 2000
  • This paper deals with a kind of the half-space interfacial problem in time domain, requiring the calculation of the horizontal electric field generated by a tiny impulsive current source located horizontally at the interface between an isotropic upper half-space and a uniaxially anisotropic lower half-space. With the Cagniard-de-Hoop method adapted for our interfacial case, we obtain the explicit-form solution for this electric field. We also investigate the impulse radiation in the radial direction. The impulse components of Dirac $\delta$-function type in transient waveforms are important for the understanding of the interfacial far-field characteristics. The uniaxial case is a generalization of the isotropic one, and the explicit solutions of the uniaxial problem in this paper reduce to the solutions of the isotropic problem if the anisotropy is removed.

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