• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1491-1501
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• 2018

Polynomials with all the coefficients in {0, 1} and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in {-1, 1} are called Littlewood polynomials. By exploiting an algorithm developed earlier, we determine the set of Littlewood polynomials of degree at most 12 which divide Newman polynomials. Moreover, we show that every Newman quadrinomial $X^a+X^b+X^c+1$, 15 > a > b > c > 0, has a Littlewood multiple of smallest possible degree which can be as large as 32765.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.6 no.1
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• pp.88-93
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• 1994

This study is concerned with an analytic derivation of the probability density function applicable for wave heights in finite water depth using two different methods. As the first method of the study, a probability density function is developed by applying a series of polynomials which is orthogonal with respect to Rayleigh probability density function. The newly derived probability density function is compared with the histogram constructed from wave data obtained in finite water depth which indicate strong non-Gaussian characteristics. Although the probability density represents the histogram very well. it has negative density at large values. Although the magnitude of the negative density is small. it negates the use of the distribution function fer estimating extreme values. As the second method of the study, a probability density function of wave height is developed by applying the maximum entropy method. The probability density function thusly derived agrees very well with the wave height distribution in shallow water, and appears to be useful in estimating extreme values and statistical properties of wave heights in finite water depth. However, a functional relationship between the probability distribution and the non-Gaussian characteristics of the data cannot be obtained by applying the maximum entropy method.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.16 no.4
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• pp.419-429
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• 2003

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of thick, hyperboloidal shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u/sub r/, u/sub θ/, u/sub z/ in the radial, circumferential, and axial directions, respectively, we taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential(strain) and kinetic energies of the hyperboloidal shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the hyperboloidal shells of revolution. Numerical results are tabulated for eighteen configurations of completely free hyperboloidal shells of revolution having two different shell thickness ratios, three variant axis ratios, and three types of shell height ratios. Poisson's ratio (ν) is fixed at 0.3. Comparisons we made among the frequencies for these hyperboloidal shells and ones which ate cylindrical or nearly cylindrical( small meridional curvature. ) The method is applicable to thin hyperboloidal shells, as well as thick and very thick ones.

• Proceedings of the KSRS Conference
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• 2005.10a
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• pp.182-185
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• 2005

Satellite cameras are calibrated before launch in detail and in general, but it cannot be guaranteed that the geometry is not changing during launch and caused by thermal influence of the sun in the orbit. Modem satellite imaging systems are based on CCD-line sensors. Because of the required high sampling rate the length of used CCD-lines is limited. For reaching a sufficient swath width, some CCD-lines are combined to a longer virtual CCD-line. The images generated by the individual CCD-lines do overlap slightly and so they can be shifted in x- and y-direction in relation to a chosen reference image just based on tie points. For the alignment and difference in scale, control points are required. The resulting virtual image has only negligible errors in areas with very large difference in height caused by the difference in the location of the projection centers. Color images can be related to the joint panchromatic scenes just based on tie points. Pan-sharpened images may show only small color shifts in very mountainous areas and for moving objects. The direct sensor orientation has to be calibrated based on control points. Discrepancies in horizontal shift can only be separated from attitude discrepancies with a good three-dimensional control point distribution. For such a calibration a program based on geometric reconstruction of the sensor orientation is required. The approximations by 3D-affine transformation or direct linear transformation (DL n cannot be used. These methods do have also disadvantages for standard sensor orientation. The image orientation by geometric reconstruction can be improved by self calibration with additional parameters for the analysis and compensation of remaining systematic effects for example caused by a not linear CCD-line. The determined sensor geometry can be used for the generation? of rational polynomial coefficients, describing the sensor geometry by relations of polynomials of the ground coordinates X, Y and Z.