• Title/Summary/Keyword: Principle of Spatial Averaging

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THE EIGENVALUE PROBLEM AND A WEAKER FORM OF THE PRINCIPLE OF SPATIAL AVERAGING

  • Kwean, Hyuk-Jin
    • The Pure and Applied Mathematics
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    • v.9 no.1
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    • pp.31-37
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    • 2002
  • In this paper, we find explicitly the eigenvalues and the eigenfunctions of Laplace operator on a triangle domain with a mixed boundary condition. We also show that a weaker form of the principle of spatial averaging holds for this domain under suitable boundary condition.

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A GEOMETRIC CRITERION FOR THE WEAKER PRINCIPLE OF SPATIAL AVERAGING

  • Kwean, Hyuk-Jin
    • Communications of the Korean Mathematical Society
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    • v.14 no.2
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    • pp.337-352
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    • 1999
  • In this paper we find a geometric condition for the weaker principle of spatial averaging (PSA) for a class of polyhedral domains. Let \ulcorner be a polyhedron in R\ulcorner, n$\leq$3. If all dihedral angles of \ulcorner are submultiples of $\pi$, then there exists a parallelopiped \ulcorner generated by n linearily independent vectors {\ulcorner}\ulcorner in R\ulcorner containing \ulcorner so that solutions of $\Delta$u+λu=0 in \ulcorner with either the boundary condition u=0 or ∂u/∂n=0 are expressed by linear combinations of those of $\Delta$u+λn=0 in \ulcorner with periodic boundary condition. Moreover, if {\ulcorner}\ulcorner satisfies rational condition, we guarantee the weaker PSA for the domain \ulcorner.

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AN EXISTENCE OF THE INERTIAL MANIFOLD FOR NEW DOMAINS

  • Kwean, Hyuk-Jin
    • Journal of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.693-707
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    • 1996
  • Consider a specific class of scalar-valued reaction diffusion equations of the form $$ (1.1) u_t = \nu\Delta u + f(u), u \in R $$ where $\nu$ < 0 is viscosity parameter and $f : R \to R$ is sufficiently smooth.

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Precision Surface Profiling of Lens Molds using a Non-contact Displacement Sensor (비접촉 변위센서를 이용한 초소형렌즈 정밀금형 형상측정)

  • Kang, Seung-Hoon;Jang, Dae-Yoon;Lee, Joohyung
    • Journal of the Korean Society of Manufacturing Process Engineers
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    • v.19 no.2
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    • pp.69-74
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    • 2020
  • In this study, we proposed a method for surface profiling aspheric lens molds using a precision displacement sensor with a spatial scanning mechanism. The precision displacement sensor is based on the confocal principle using a broadband light source, providing a 10 nm resolution over a 0.3 mm measurable range. The precision of the sensor, depending on surface slope, was evaluated via Allan deviation analysis. We then developed an automatic surface profiling system by measuring the cross-sectional profile of a lens mold. The precision of the sensor at the flat surface was 10 nm at 10 ms averaging time, while 200 ms averaging time was needed for identical precision at the steepest slope at 25 deg. When we compared the measurement result of the lens mold to a commercial surface profiler, we found that the accuracy of the developed system was less than 90 nm (in terms of 3 sigmas of error) between the two results.

Nonlinear free vibration of heated corrugated annular plates with a centric rigid mass

  • Wang, Yong-Gang;Li, Dan;Feng, Ze-Jun
    • Structural Engineering and Mechanics
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    • v.34 no.4
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    • pp.491-505
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    • 2010
  • A computational analysis of the nonlinear free vibration of corrugated annular plates with shallow sinusoidal corrugations under uniformly static ambient temperature is examined. The governing equations based on Hamilton's principle and nonlinear bending theory of thin shallow shell are established for a corrugated plate with a concentric rigid mass at the center and rotational springs at the outer edges. A simple harmonic function in time is assumed and the time variable is eliminated from partial differential governing equations using the Kantorovich averaging procedure. The resulting ordinary equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude for nonlinear vibration of heated corrugated annular plates are obtained. Several numerical results are presented in both tabular and graphical forms, which demonstrate the accuracy of present method and illustrate the amplitude frequency dependence for the plate under such parameters as ambient temperature, plate geometry, rigid mass and elastic constrain.