• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.345-351
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• 2007

Using more precise majorizing sequences we provide a finer convergence analysis than before [1], [7] of Newton's method in Riemannian manifolds with the following advantages: weaker hypotheses, finer error bounds on the distances involved and a more precise information on the location of the singularity of the vector field.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.17-53
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• 2014

Let ${\gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({\gamma})$ be the chain component of X which contains ${\gamma}$. We say that $C_X({\gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of X such that for any $Y{\in}\mathcal{U}$, $C_Y({\gamma}_Y)$ is shadowable for $Y_t$, where ${\gamma}_Y$ denotes the continuation of ${\gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({\gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({\gamma})$ has no non-hyperbolic singularity.

• Journal of the Optical Society of Korea
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• v.19 no.4
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• pp.397-402
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• 2015

In this paper, we propose a method used to measure the polarization state of a field with multiple polarization singularities (PSs), such as an array of PSs and PSs in vector speckle. A checkerboard is used to construct characteristic points, which makes it possible to calibrate the mismatches between pictures obtained during the rotation of a quarter-wave plate and a polarizer. Using this method a field with an array of PSs is measured. The experimental result is compared with the numerical simulation. We also carry out some data analysis. The comparison and analysis emphasize the necessity and feasibility of this method to measure the PSs.