• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.461-473
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• 2022

In this paper, we study differential equations arising from the generating functions of the geometric polynomials. We give explicit identities for the geometric polynomials. Finally, we investigate the zeros of the geometric polynomials by using computer.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.1-20
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• 2016

The general number field sieve (GNFS) is asymptotically the fastest known factoring algorithm. One of the most important steps of GNFS is to select a good polynomial pair. A standard way of polynomial selection (being used in factoring RSA challenge numbers) is to select a nonlinear polynomial for algebraic sieving and a linear polynomial for rational sieving. There is another method called a nonlinear method which selects two polynomials of the same degree greater than one. In this paper, we generalize Montgomery's method [12] using geometric progression (GP) (mod N) to construct a pair of nonlinear polynomials. We also introduce GP of length d + k with $1{\leq}k{\leq}d-1$ and show that we can construct polynomials of degree d having common root (mod N), where the number of such polynomials and the size of the coefficients can be precisely determined.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.641-646
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• 2004

A convex combination of two products with same degree of finitely many finite geometric series with each having even degree does not always have all its zeros on the unit circle. However, in this paper, we show that a polynomial obtained by just adding a finite geometric series multiplied by a large constant to such a convex combination has all its zeros on the unit circle.

• Structural Engineering and Mechanics
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• v.17 no.6
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• pp.751-766
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• 2004

This paper extends the use of the hierarchic degenerated shell element to geometric non-linear analysis of composite laminated skew plates by the p-version of the finite element method. For the geometric non-linear analysis, the total Lagrangian formulation is adopted with moderately large displacement and small strain being accounted for in the sense of von Karman hypothesis. The present model is based on equivalent-single layer laminate theory with the first order shear deformation including a shear correction factor of 5/6. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. A wide variety of linear and non-linear results obtained by the p-version finite element model are presented for the laminated skew plates as well as laminated square plates. A numerical analysis is made to illustrate the influence of the geometric non-linear effect on the transverse deflections and the stresses with respect to width/depth ratio (a/h), skew angle (${\beta}$), and stacking sequence of layers. The present results are in good agreement with the results in literatures.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.10C
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• pp.614-620
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• 2011

RSA, the most commonly used public-key cryptosystem, is based on the difficulty of factoring very large integers. The fastest known factoring algorithm is the Number Field Sieve(NFS). NFS first chooses two polynomials having common root modulo N and consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root, of which the most time consuming step is the Sieving step. However, in recent years, the importance of the Polynomial Selection step has been studied widely, because one can save a lot of time and memory in sieving and matrix step if one chooses optimal polynomial for NFS. One of the ideal ways of choosing sieving polynomial is to choose two polynomials with same degree. Montgomery proposed the method of selecting two (nonlinear) quadratic sieving polynomials. We proposed two cubic polynomials using 5-term geometric progression.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.461-473
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• 2007

For integral self-reciprocal polynomials P(z) and Q(z) with all zeros lying on the unit circle, does there exist integral self-reciprocal polynomial $G_r(z)$ depending on r such that for any r, $0{\leq}r{\leq}1$, all zeros of $G_r(z)$ lie on the unit circle and $G_0(z)$ = P(z), $G_1(z)$ = Q(z)? We study this question by providing examples. An example answers some interesting questions. Another example relates to the study of convex combination of two polynomials. From this example, we deduce the study of the sum of certain two products of finite geometric series.

• Korean Journal of Geomatics
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• v.3 no.1
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• pp.53-57
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• 2003

As the Korea Multi-Purpose Satellite-I (KOMPSAT-1) satellite can roll tilt up to $\pm$45$^{\circ}$, we have analyzed some KOMPSAT-1 EOC images taken at different tilt angles for this study. The required ground coordinates for bundle adjustment and geometric accuracy are obtained from the digital map produced by the National Geography Institution, at a scale of 1:5,000. Followings are the steps taken for the tilting angle of KOMPSAT-1 to be present in the evaluation of geometric accuracy of each different stereo image data: Firstly, as the tilting angle is different in each image, the characteristic of satellite dynamic must be determined by the sensor modeling. Then the best sensor modeling equation should be determined. The result of this research, the difference between the RMSE values of individual stereo images is mainly due to quality of image and ground coordinates instead of tilt angle. The bundle adjustment using three KOMPSAT-1 stereo pairs, first degree of polynomials for modeling the satellite position, were sufficient.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2006.04a
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• pp.623-630
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• 2006

In this paper, a geometric nonlinear analysis procedure of beam-column element including multi-noded cable element is presented. For this, first a stiffness matrix about beam-column element which considers the second effect of initial force supposing the curved shape at each time step with Hermitian polynomials as the shape function is derived and second, tangent stiffness matrix about multi-noded cable element being too. To verify geometric nonlinearity of this newly developed multi-noded cable-truss element, IPS(Innovative Prestressed Support) system using this theory is analysed by geometric nonlinear method and the results are compared with those by linear analysis.

• Korean Journal of Remote Sensing
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• v.12 no.1
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• pp.17-25
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• 1996

This paper presents a GCP marking technique using scanned maps. Twelve maps with a scale of 1:250, 000 were scanned and stored as raster images. The distortion factors of scanned maps were modelled by polynomials. The coefficients of the polynomials were determined by a least squre fit. This technique does not require a bulky and expensive digitizing table so that it is suitable for a low-cost pre-processing system. The GCP marking using this technique showed a sufficient accuracy for KITSAT1, 2 narrow camera images.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.7-10
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• 2009

With the introduction of a new parameter $n{\geq}1$, Kim generalized an upper bound for the exponential function that implies the inequality between the arithmetic and geometric means. By a change of variable, this generalization is equivalent to exp $(\frac{n(x-1)}{n+x-1})\;\leq\;\frac{n-1+x^n}{n}$ for real ${n}\;{\geq}\;1$ and x > 0. In this paper, we show that this inequality is true for real x > 1 - n provided that n is an even integer.