• 정보보호학회논문지
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• 제15권6호
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• pp.105-110
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• 2005

유한체에서 정의된 선형 다항식의 역원은 함수 거 일반화로 볼 수 있으므로, 암호학적 응용에서 유용한 부울 하수를 설계하는 데 좋은 후보가 될 수 있다 특히, Crypto 2001에서는 선형 다항식 및 선형 부호를 이용하여 큰 대수적 차수를 가지는 resilient 함수를 설계하는 방법이 제안되었다. 그러나 Crypto 2001에서 대수적 차수를 분석한 결과에 오류가 있었으며, 본 논문에서 정확한 대수적 차수를 제시한다.

• 대한수학회논문집
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• 제37권4호
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• pp.1259-1267
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• 2022

In this paper we present a full circle approximation method using parametric polynomial curves with algebraic coefficients which are curvature continuous at both endpoints. Our method yields the n-th degree parametric polynomial curves which have a total number of 2n contacts with the full circle at both endpoints and the midpoint. The parametric polynomial approximants have algebraic coefficients involving rational numbers and radicals for degree higher than four. We obtain the exact Hausdorff distances between the circle and the approximation curves.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.135-143
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• 2011

As a zero set of some multivariate splines, the piecewise algebraic variety is a kind of generalization of the classical algebraic variety. This paper presents an algorithm for isolating real roots of the zero-dimensional piecewise algebraic variety which is based on interval evaluation and the interval zeros of univariate interval polynomials in Bernstein form. An example is provided to show the proposed algorithm is effective.

• 대한수학회보
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• 제59권1호
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• pp.45-72
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• 2022

In this work, we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with piecewise smooth boundary without cusps of the complex plane. Also, estimates are given on the whole complex plane.

• 대한수학회보
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• 제32권1호
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• pp.73-83
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• 1995

Let G be a real algebraic group. In this paper we consider two isomorphisms of real algebraic G varieties and real algebraic G vector bundles.

• Kyungpook Mathematical Journal
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• 제28권2호
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• pp.185-196
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• 1988

We study existence of polynomial integrating factors and solutions F(x, y)=c of first order nonlinear differential equations. We characterize the homogeneous case, and give algorithms for finding existence of and a basis for polynomial solutions of linear difference and differential equations and rational solutions or linear differential equations with polynomial coefficients. We relate singularities to nature of the solution. Solution of differential equations in closed form to some degree might be called more an art than a science: The investigator can try a number of methods and for a number of classes of equations these methods always work. In particular integrating factors are tricky to find. An analogous but simpler situation exists for integrating inclosed form, where for instance there exists a criterion for when an exponential integral can be found in closed form. In this paper we make a beginning in several directions on these problems, for 2 variable ordinary differential equations. The case of exact differentials reduces immediately to quadrature. The next step is perhaps that of a polynomial integrating factor, our main study. Here we are able to provide necessary conditions based on related homogeneous equations which probably suffice to decide existence in most cases. As part of our investigations we provide complete algorithms for existence of and finding a basis for polynomial solutions of linear differential and difference equations with polynomial coefficients, also rational solutions for such differential equations. Our goal would be a method for decidability of whether any differential equation Mdx+Mdy=0 with polynomial M, N has algebraic solutions(or an undecidability proof). We reduce the question of all solutions algebraic to singularities but have not yet found a definite procedure to find their type. We begin with general results on the set of all polynomial solutions and integrating factors. Consider a differential equation Mdx+Ndy where M, N are nonreal polynomials in x, y with no common factor. When does there exist an integrating factor u which is (i) polynomial (ii) rational? In case (i) the solution F(x, y)=c will be a polynomial. We assume all functions here are complex analytic polynomial in some open set.

• 한국정밀공학회지
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• 제21권7호
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• pp.76-82
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• 2004

The solutions of neighboring optimal control are typically obtained using the sweep method or transition matrices. Due to the numerical integration, however, the gain matrix can become infinite as time go to final one in the transition matrices, and the Riccati solution can become infinite when the final time free. To overcome these disadvantages, this paper proposes the pseudospectral Legendre method which is to first discreteize the linear boundary value problem using the global orthogonal polynomial, then transforms into an algebraic equations. Because this method is not necessary to take any integration of transition matrix or Riccati equation, it can be usefully used in real-time operation. Finally, its performance is verified by the numerical example for the space vehicle's orbit transfer.

• 대한수학회보
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• 제53권1호
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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.

• 한국정보과학회논문지:소프트웨어및응용
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• 제27권1호
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• pp.50-61
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• 2000

최근 원추곡선에 기반한 스테레오 비젼에 대한 연구가 주목을 받고 있는데, 이는 원추곡선이 행렬표현, 대응관계설정의 용이성, 그리고 실세계에서 쉽게 찾을 수 있다는 좋은 성질을 갖는다는 점에서 당연한 현상이라 여겨진다. 하지만, 일반적인 고차의 대수곡선에 대한 확장은 아직 성공적으로 이루어지지 못하고 있는 실정이다. 기약인 대수곡선 (irreducible algebraic curve)은 실세계에서 많지 않지만, 직선과 원추곡선은 무수히 많고, 따라서 이들의 곱으로 주어지는 높은 차수의 대수곡선도 무수히 많다. 본고에서는 2이상의 임의의 차수를 가지는 대수곡선을 calibration된 두 대의 카메라를 가지고 스테레오 문제를 푼다. 대응관계설정과 복원, 두 가지 문제 모두에 대한 closed form solution을 제시한다. $f_1,\;f_2,\;{\pi}$를 각각 두 이미지 곡선, 공간상의 평면이라 하고, $VC_P(g)$를 평면곡선 g와 점 P로 만들어지는 원추곡선이라 하면, $VC_{O1}(f_1)\;=\;VC_{O1}(VC_{O2}(f_2)\;∩\;{\pi})$ 의 관계를 이용하여 미지수인 평면 ${\pi}$의 계수들, $d_1,\;d_2,\;d_3$에 대한 다항 방정식들을 얻을 수 있다. 약간의 변형을 통하여 $d_1$에 대한 다항 방정식을 얻을 수 있고, 이 방정식의 유일한 양수해는 나머지 과정에서 매우 중요한 역할을 한다. 그 이후에는 $O(n^2)$개의 일변수 다항식에 대한 계산만으로 모든 스테레오 문제를 해결한다. 이는 과거의 여러 개의 다변수 다항식의 공통근을 구해야 했던 방법에 비교된다. synthetic 데이터와 실제 이미지에 대한 실험은 우리의 알고리듬이 옳음을 보여준다.

• 대한수학회논문집
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• 제31권4호
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• pp.667-682
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• 2016

In this article we show a recursive method to compute the coefficients of the minimal polynomial of cos($2{\pi}/n$) explicitly for $n{\geq}3$. The recursion is not on n but on the coefficient index. Namely, for a given n, we show how to compute ei of the minimal polynomial ${\sum_{i=0}^{d}}(-1)^ie_ix^{d-i}$ for $i{\geq}2$ with initial data $e_0=1$, $e_1={\mu}(n)/2$, where ${\mu}(n)$ is the $M{\ddot{o}}bius$ function.