• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.159-169
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• 2005

An algorithmic approach to degree reduction of rational Bezier curves is presented. The algorithms are based on the degree reduction of polynomial Bezier curves. The method is introduced with the following steps: (a) convert the rational Bezier curve to polynomial Bezier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bezier curve, (c) determine weights of degree reduced curve, (d) convert the Bezier curve obtained through step (b) to rational Bezier curve with weights in step (c).

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.279-284
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• 2008

In this paper, we present the constrained Jacobi polynomial which is equal to the constrained Chebyshev polynomial up to constant multiplication. For degree n=4, 5, we find the constrained Jacobi polynomial, and for $n{\geq}6$, we present the normalized constrained Jacobi polynomial which is similar to the constrained Chebyshev polynomial.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.15 no.6
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• pp.105-110
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• 2005

The linearized polynomial fan be regarded as a generalization of the identity function so that the inverse of the linearized polynomial is a generalization of e inverse function. Since the inverse function has so many good cryptographic properties, the inverse of the linearized polynomial is also a candidate of good Boolean functions. In particular, a construction method of vector resilient functions with high algebraic degree was proposed at Crypto 2001. But the analysis about the algebraic degree of the inverse of the linearized Polynomial. Hence we correct the inexact result and give the exact maximal algebraic degree.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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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 the Korean Mathematical Society
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• v.33 no.2
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• pp.423-438
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• 1996

Let L be a linear functional on the vector space of polynomials in x. Let $\omega(x)$ be a polynomial in x of degree d, for some positive integer d. We consider two sets of polynomials, ${R_n (x)}_{n \geq 0}, {S_n(x)}_{n \geq 0}$, such that $R_n(x)$ is a polynomial in x of degree n and $S_n(x)$ is a polynomial in $\omega(x)$ of degree n. (So $S_n(x)$ is a polynomial in x of degree dn.)

• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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• pp.183-198
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• 2009

In this paper, we derive a characterization of orthonormal balanced multiwavelets of order p in terms of the continuous moments of the multiscaling function $\phi$. As a result, the continuous moments satisfy the discrete polynomial preserving properties of order p (or degree p - 1) for orthonormal balanced multiwavelets. We derive polynomial reproduction formula of degree p - 1 in terms of continuous moments for orthonormal balanced multiwavelets of order p. Balancing of order p implies that the series of scaling functions with the discrete-time monomials as expansion coefficients is a polynomial of degree p - 1. We derive an algorithm for computing the polynomial of degree p - 1.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.2
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• pp.99-106
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• 2005

We present a relation between the orthogonality of the constrained Legendre polynomials over the triangular domain and the BB ($B{\acute{e}zier}\;-Bernstein$) coefficients of the polynomials using the equivalence of orthogonal complements. Using it we also show that the best constrained degree reduction of polynomials in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.675-678
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• 2006

We show that every real polynomial of degree n can be represented by the sum of two real polynomials of degree n, each having only real zeros. As an example of this, we consider a real polynomial with even degree whose all zeros lie on the imaginary axes except the origin.

• Journal of the Chungcheong Mathematical Society
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• v.16 no.1
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• pp.7-14
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• 2003

Let $P_c(z)=z^n+c$ be a complex polynomial with an integer $n{\geq}2$. We derive a criterion that the critical orbit of $P_c$ escapes to infinity and investigate its applications to the degree-n bifurcation set. The intersection of the degree-n bifurcation set with the real line as well as with a typical symmetric axis is explicitly written as a function of n. A well-defined escape-time algorithm is also included for the improved construction of the degree-n bifurcation set.

• 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.