• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.473-485
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• 2010

We consider the differential equation $$(x^2\;-\;1)u_{xx}\;+\;2xyu_{xy}\;+\;(y^2\;-\;1)u_{yy}\;+\;gxu_x\;+\;gyu_y\;=\;\lambda_nu,\;(*)$$ where $\lambda_n\;=\;n(n\;+\;9\;-\;1)$. We show that the differential equation (*) has a polynomial set as solutions if $g\;{\neq}\;-1$, -3, -5, $\cdots$. Also, we construct an orthogonalizing distributional weight for g < 1 and $g\;{\neq}\;1$, 0, -1, $\cdots$ by regularizing a one-dimensional integral with a singularity on the endpoint of the interval.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.95-113
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• 2018

A mathematical knot is an embedded circle in ${\mathbb{R}}^3$. A fundamental problem in knot theory is classifying knots up to its numbers of crossing points. Knots are often distinguished by using a knot invariant, a quantity which is the same for equivalent knots. Knot polynomials are one of well known knot invariants. In 2006, J. Przytycki showed the effects of a n - move (a local change in a knot diagram) on several knot polynomials. In this paper, the authors review about knot polynomials, especially Jones polynomial, and give an alternative proof to a part of the Przytychi's result for the case n = 3 on the Jones polynomial.

• Journal of the Semiconductor & Display Technology
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• v.9 no.1
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• pp.11-16
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• 2010

Wafer Pre-Alignment is to find the center and the orientation of a wafer and to move the wafer to the desired position and orientation. In this paper, an area camera based pre-aligning method is presented that captures 8 wafer images regularly during 360 degrees rotation. From the images, wafer edge positions are extracted and used to estimate the wafer's center and orientation using least squares circle fitting. These data are utilized for the proper alignment of the wafer. For accurate alignments, camera calibration methods using high order polynomials are used for converting pixel coordinates into real-world coordinates. A complete pre-alignment system was constructed using mechanical and optical components and tested. Experimental results show that alignment of wafer center and orientation can be done with the standard deviation of 0.002 mm and 0.028 degree, respectively.

• Journal of Integrative Natural Science
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• v.10 no.1
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• pp.40-42
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• 2017

In this note, we study a generalization of a certain polynomial $z^n-{\sum\limits_{k=0}^{n-1}}a_kz^k$, where $\sum\limits_{k=0}^{n-1}a_k=1$, $a_k{\geq}0$ for each k, whose all zeros except for z=1 lie on the circle of radius 1/2 with center at the origin.

• 한국전산유체공학회:학술대회논문집
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• 2007.04a
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• pp.30-40
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• 2007

The implicit discontinuous Galerkin method for the two-dimensional Euler equations was developed on unstructured triangular meshes, which can achieve higher-order accuracy by wing hierachical basis functions based on Legendre polynomials. Numerical tests were conducted to estimate the convergence order of numerical solutions to the Ringleb flow and the supersonic vortex flow for which analytic solutions are available. And, the flows around a circle and a NACA0012 airfoil was also numerically simulated. Numerical results show that the implicit discontinuous Galerkin methods with higher-order representation of curved solid boundaries can be an efficient higher-order method to obtain very accurate numerical solutions on unstructured meshes.

• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.691-700
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• 2022

Let p(z) be a polynomial of degree n having no zero inside the unit circle. Then for 0 < r ≤ 1, the well-known inequality due to Rivlin [Amer. Math. Monthly., 67 (1960) 251-253] is $$\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}{\geq}{\(\frac{r+1}{2}\)^n}\max\limits_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$. In this paper, we generalize as well as sharpen the above inequality. Also our results not only generalize, but also sharpen some known results proved recently.