• Journal of KIISE:Software and Applications
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• v.27 no.1
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• pp.50-61
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• 2000

Recently the stereo vision based on conics has received much attention by many authors. Conics have many features such as their matrix expression, efficient correspondence checking, abundance of conical shapes in real world. Extensions to higher algebraic curves met with limited success. Although irreducible algebraic curves are rather rare in the real world, lines and conics are abundant whose products provide good examples of higher algebraic curves. We consider plane algebraic curves of an arbitrary degree $n{\geq}2$ with a fully calibrated stereo system. We present closed form solutions to both correspondence and reconstruction problems. Let $f_1,\;f_2,\;{\pi}$ be image curves and plane and $VC_P(g)$ the cone with generator (plane) curve g and vertex P. Then the relation $VC_{O1}(f_1)\;=\;VC_{O1}(VC_{O2}(f_2)\;∩\;{\pi})$ gives polynomial equations in the coefficient $d_1,\;d_2,\;d_3$ of the plane ${\pi}$. After some manipulations, we get an extremely simple polynomial equation in a single variable whose unique real positive root plays the key role. It is then followed by evaluating $O(n^2)$ polynomials of a single variable at the root. It is in contrast to the past works which usually involve a simultaneous system of multivariate polynomial equations. We checked our algorithm using synthetic as well as real world images.

• Structural Engineering and Mechanics
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• v.14 no.3
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• pp.263-285
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• 2002

This paper presents a numerical procedure for solving initial-value problems using the special functions which belong to a class of Rvachev's basis functions $R_{bf}$ based on algebraic and trigonometric polynomials. Because of infinite derivability of these functions, derivatives of all orders, required by differential equation of the problem and initial conditions, are used directly in the numerical procedure. The accuracy and stability of the proposed numerical procedure are proved on an example of a single degree of freedom system. Critical time step was also determined. An algorithm for solving multiple degree of freedom systems by the collocation method was developed. Numerical results obtained by $R_{bf}$ functions are compared with exact solutions and results obtained by the most commonly used numerical procedures for solving initial-value problems.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.299-309
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• 2013

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method (FChFD). The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. We tested this technique to solve numerically fractional BVPs. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The fractional derivatives are presented in terms of Caputo sense. The application of the method to fractional BVPs leads to algebraic systems which can be solved by an appropriate method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

• Proceedings of the Korea Concrete Institute Conference
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• 1998.10a
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• pp.412-417
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• 1998

This paper provides accurate flexural vibration solutions for thick (Mindlin) sectorial plates. A Ritz method is employed which incorporates a complete set of admissible algebraic-trigonometric polynomials in conjunction with an admissible set of Mindlin “corner functions". These corner functions model the singular vibratory moments and shear forces, which simultaneously exist at the vertex of corner angle exceeding 180$^{\circ}$. The first set guarantees convergence to the exact frequencies as sufficient terms are taken. The second set represents the corner singularities, and accelerates convergence substantially. Numerical results are obtained for completely free sectorial plates. Accurate frequencies are presented for a wide spectrum of vertex angles (90$^{\circ}$, 180$^{\circ}$, 270$^{\circ}$, 300$^{\circ}$, 330$^{\circ}$, 350$^{\circ}$, 35 5$^{\circ}$,and 359$^{\circ}$)and thickness ratios.tios.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.17-24
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• 2002

It will be described the discovery of fundamental algebras such as complex numbers and the quaternions. Cardano(1539) was the first to introduce special types of complex numbers such as 5$\pm$$\sqrt{-15}$. Girald called the number a$\pm$$\sqrt{-b}$ solutions impossible. The term imaginary numbers was introduced by Descartes(1629) in “Discours la methode, La geometrie.” Euler knew the geometrical representation of complex numbers by points in a plane. Geometrical definitions of the addition and multiplication of complex numbers conceiving as directed line segments in a plane were given by Gauss in 1831. The expression “complex numbers” seems to be Gauss. Hamilton(1843) defined the complex numbers as paire of real numbers subject to conventional rules of addition and multiplication. Cauchy(1874) interpreted the complex numbers as residue classes of polynomials in R[x] modulo $x^2$＋1. Sophus Lie(1880) introduced commutators [a, b] by the way expressing infinitesimal transformation as differential operations. In this paper, it will be studied general quaternion algebras to finding of algebraic structure in Algebras.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.14 no.4
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• pp.336-343
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• 2004

This paper reports the first known free vibration data for thin rectangular plates with V-notches. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained, and (2) corner functions which account for the bending moment singularities at the sharp reentrant corner of the Y-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondirectional frequencies for rectangular plates having the V-notch. In this paper, accurate frequencies and normalized contours of vibratory transverse displacement are presented for various notched plates, so that the effect of corner stress singularities may be understood.

• Nuclear Engineering and Technology
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• v.31 no.2
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• pp.214-225
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• 1999

An accurate method is presented for flexural vibrations of sectorial plates having simply supported-free and free-free radial edges, when the circular edge is either clamped or simply supported. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets consist of : (1) mathematically complete algebraic-trigonometric polynomials which gurantee convergence to exact frequencies as sufficient terms are retained, and (2) comer functions which account for the bending moment singularities at re-entrant comer of the radial edges having arbitrary edge conditions. Accurate (at least four significant figures) frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of corner angles [90$^{\circ}$, 180$^{\circ}$(semi-circular), 270$^{\circ}$, 300$^{\circ}$, 330$^{\circ}$, 350$^{\circ}$, 355$^{\circ}$, 360$^{\circ}$ (complete circular)] causing a re-entrant comer of the radial edges. Future solutions drawn from alternative numerical procedures and finite element techniques may be compared with these accurate results.

• Advances in nano research
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• v.6 no.3
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• pp.257-278
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• 2018

Vibration of axially functionally graded nano-rods and beams is investigated. It is assumed that the material properties change along the rod and beam length. The Ritz method with algebraic polynomials is used in the formulation of the problems. Stress gradient elasticity theory is utilized in order to include the nonlocal effects. Frequencies are obtained for different boundary conditions, geometrical and material properties. Nonlocal parameter is assumed as changing linearly or quadratically along the length of the nanostructure. Frequencies are compared to constant nonlocal parameter cases and considerable differences are observed between constant and variable nonlocal parameter cases. Mode shapes in various cases are depicted in order to explain the effects of axial grading.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.717-735
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• 2004

A problem raised by Selfridge and solved by Pomerance asks to find the pairs (a, b) of natural numbers for which $2^a\;-\;2^b$ divides $n^a\;-\;n^b$ for all integers n. Vajaitu and one of the authors have obtained a generalization which concerns elements ${\alpha}_1,\;{\cdots},\;{{\alpha}_{\kappa}}\;and\;{\beta}$ in the ring of integers A of a number field for which ${\Sigma{\kappa}{i=1}}{\alpha}_i{\beta}^{{\alpha}i}\;divides\;{\Sigma{\kappa}{i=1}}{\alpha}_i{z^{{\alpha}i}}\;for\;any\;z\;{\in}\;A$. Here we obtain a further generalization, proving the corresponding finiteness results in a multidimensional setting.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.459-479
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• 2020

In the last two decades, codes over noncommutative rings have been one of the main trends in coding theory. Due to the fact that noncommutativity brings many challenging problems in its nature, still there are many open problems to be addressed. In 2015, generator polynomial matrices and parity-check polynomial matrices of generalized quasi-cyclic (GQC) codes were investigated by Matsui. We extended these results to the noncommutative case. Exploring the dual structures of skew constacyclic codes, we present a direct way of obtaining parity-check polynomials of skew multi-twisted codes in terms of their generators. Further, we lay out the algebraic structures of skew multipolycyclic codes and their duals and we give some examples to illustrate the theorems.