• Title/Summary/Keyword: Curvature measure

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Curvature Measurement Using Fiber Bragg Gratings (파이버 브래그 격자를 이용한 곡률 측정)

  • Jung, Jin-Ho;Lee, Jong-Youn
    • Journal of the Korea Academia-Industrial cooperation Society
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    • v.9 no.4
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    • pp.1081-1085
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    • 2008
  • To measure the curvature, in this paper, we investigate an optical curvature sensor based on the fiber Bragg gratings. We observed the variation of the Bragg resonant wavelength shift to measure the curvature change. From the experimental results, we knew that the Brags resonant wavelength shift was lineally increased with the increase of the curvature from $0\;m^{-1}$ to $10\;m^{-1}$. It's slope is about $8.8\;pm/m^{-1}$. On the other hand, the spectral reflection decreased with the increase of the curvature.

THE HARDY TYPE INEQUALITY ON METRIC MEASURE SPACES

  • Du, Feng;Mao, Jing;Wang, Qiaoling;Wu, Chuanxi
    • Journal of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1359-1380
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    • 2018
  • In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.

Effects of curvature on leverage in nonlinear regression

  • Kahng, Myung-Wook
    • Journal of the Korean Data and Information Science Society
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    • v.20 no.5
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    • pp.913-917
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    • 2009
  • The measures of leverage in linear regression has been extended to nonlinear regression models. We consider several curvature measures of nonlinearity in an estimation situation. The relationship between measures of leverage and statistical curvature are explored in nonlinear regression models. The circumstances under which the Jacobian leverage reduces to a tangent plane leverage are discussed in connection with the effective residual curvature of the nonlinear model.

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HYPERSURFACES WITH PRESCRIBED MEAN CURVATURE IN MEASURE METRIC SPACE

  • Zhengmao Chen
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.1085-1100
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    • 2023
  • For any given function f, we focus on the so-called prescribed mean curvature problem for the measure e-f(|x|2)dx provided thate-f(|x|2) ∈ L1(ℝn+1). More precisely, we prove that there exists a smooth hypersurface M whose metric is ds2 = dρ2 + ρ2d𝜉2 and whose mean curvature function is ${\frac{1}{n}}(\frac{u^p}{{\rho}^{\beta}})e^{f({\rho}^2)}{\psi}(\xi)$ for any given real constants p, β and functions f and ψ where u and ρ are the support function and radial function of M, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere 𝕊n, $${\sum_{i,j}}({{\delta}_{ij}-{\frac{{\rho}_i{\rho}_j}{{\rho}^2+|{\nabla}{\rho}|^2}})(-{\rho}ji+{\frac{2}{{\rho}}}{\rho}j{\rho}i+{\rho}{\delta}_{ji})={\psi}{\frac{{\rho}^{2p+2-n-{\beta}}e^{f({\rho}^2)}}{({\rho}^2+|{\nabla}{\rho}|^2)^{\frac{p}{2}}}}$$ under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)={\frac{t}{2}}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

A Study on the Surface Deflection in Rectangular Embossing Considering Planar Anisotropy (평면이방성을 고려한 사각엠보싱 공정의 미세면굴곡에 대한 연구)

  • Kim, J.H.;Chung, W.J.
    • Transactions of Materials Processing
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    • v.22 no.6
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    • pp.310-316
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    • 2013
  • Recently, numerical predictions of surface deflection based on curvature analysis have been developed. In the current study, a measure of surface deflection is proposed as the maximum variation of curvature difference between the panel and the tool in order to account for surfaces that have high curvature. The current study focused on the assessment of accuracy for the surface deflection prediction with the consideration of planar anisotropy. As an example, a shallow rectangular drawn part with rectangular embossing was considered. In terms of the proposed surface deflection measure, the maximum variation of curvature difference, the prediction with a planar anisotropic model shows better correspondence with experiment than the one using a normal anisotropic model.

THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATURE

  • Zhou, Jiazu;Chen, Fangwei
    • Journal of the Korean Mathematical Society
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    • v.44 no.6
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    • pp.1363-1372
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    • 2007
  • We investigate the containment measure of one domain to contain in another domain in a plane $X^{\kappa}$ of constant curvature. We obtain some Bonnesen-type inequalities involving the area, length, radius of the inscribed and the circumscribed disc of a domain D in $X^{\kappa}$.

COMPARISON THEOREMS FOR THE VOLUMES OF TUBES ABOUT METRIC BALLS IN CAT(𝜿)-SPACES

  • Lee, Doohann;Kim, Yong-Il
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.3
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    • pp.457-467
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    • 2011
  • In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space ${\mathbb{R}}^n$, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

Effect of the Nozzle Curvature on Critical Flows (임계노즐 유동에 미치는 노즐 곡률의 영향)

  • Kim, Jae-Hyung;Kim, Heuy-Dong;Park, Kyung-Am
    • 유체기계공업학회:학술대회논문집
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    • 2002.12a
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    • pp.331-336
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    • 2002
  • Recently the critical nozzles with small diameter are being extensively used to measure mass flow in a variety of industrial fields and these have different configurations depending on operation condition and working gas. The curvature radius of the critical nozzle throat is one of the most important configuration factors promising a high reliability of the critical nozzle. In the present study, computations using the axisymmetric, compressible, Navier-Stokes equations are carried out to investigate the effect of the nozzle curvature on critical flows. The diameter of the critical nozzle employed is D=0.3mm and the radius of curvature of the critical nozzle throat is varied in the range from 1D to 3D. It is found that the discharge coefficient is very sensitive to the curvature radius(R) of critical nozzle, leading to the peak discharge coefficient at R = 2.0D and 2.5D, and that the critical pressure ratio increases with the curvature radius.

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Prediction of Residual Stress Distribution in Multi-Stacked Thin Film by Curvature Measurement and Iterative FEA

  • Choi Hyeon Chang;Park Jun Hyub
    • Journal of Mechanical Science and Technology
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    • v.19 no.5
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    • pp.1065-1071
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    • 2005
  • In this study, residual stress distribution in multi-stacked film by MEMS (Micro-Electro Mechanical System) process is predicted using Finite Element method (FEM). We evelop a finite element program for residual stress analysis (RESA) in multi-stacked film. The RESA predicts the distribution of residual stress field in multi-stacked film. Curvatures of multi­stacked film and single layers which consist of the multi-stacked film are used as the input to the RESA. To measure those curvatures is easier than to measure a distribution of residual stress. To verify the RESA, mean stresses and stress gradients of single and multi layers are measured. The mean stresses are calculated from curvatures of deposited wafer by using Stoney's equation. The stress gradients are calculated from the vertical deflection at the end of cantilever beam. To measure the mean stress of each layer in multi-stacked film, we measure the curvature of wafer with the left film after etching layer by layer in multi-stacked film.