• Nonlinear Functional Analysis and Applications
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• 제27권4호
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• pp.797-805
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• 2022

Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ P_n, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

• Nonlinear Functional Analysis and Applications
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• 제28권2호
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• pp.421-437
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• 2023

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then Malik [12] obtained the following inequality: $${_{max \atop {\mid}z{\mid}=1}{\mid}p{\prime}(z){\mid}{\leq}{\frac{n}{1+k}}{_{max \atop {\mid}z{\mid}=1}{\mid}p(z){\mid}.$$ In this paper, we shall first improve as well as generalize the above inequality. Further, we also improve the bounds of two known inequalities obtained by Govil et al. [8].

• 대한수학회지
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• 제45권3호
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• pp.835-840
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• 2008

The main result of this paper shows that every reciprocal Littlewood polynomial, one with {-1, 1} coefficients, of odd degree at least 7 has at least five unimodular roots, and every reciprocal Little-wood polynomial of even degree at least 14 has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in $$A_N=\{[{X^d+ \sum\limits^{d-1}_{k=0} a_k\;X^k{\in} \mathbb{Z}[X]\;:\;a_k={\pm}N,\;0{\leqslant}k{\leqslant}d-1}\}$$ for positive integer $N{\geqslant}2$.

• 한국정보통신학회논문지
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• 제12권7호
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• pp.1209-1217
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• 2008

본 논문에서는 $GF(2^m)$ 상에서 표준기저를 사용한 두 다항식의 곱셈을 비트-병렬로 실현하는 새로운 형태의 비트-병렬 곱셈기를 제안하였다. 곱셈기의 구성에 앞서, 피승수 다항식과 기약다항식의 곱셈을 병렬로 수행 한 후 승수 다항식의 한 계수와 비트-병렬로 곱셈하여 결과를 생성하는 VCG를 구성하였다. VCG의 기본 셀은 2개의 AND 게이트와 2개의 XOR 게이트로 구성되며, 이들로부터 두 다항식의 비트-병렬 곱셈을 수행하여 곱셈 결과를 얻도록 하였다. 이러한 과정을 확장하여 m에 대한 일반화된 회로의 설계를 보였으며, 간단한 형태의 곱셈회로 구성의 예를 $GF(2^4)$를 통해 보였다. 또한 제시한 곱셈기는 PSpice 시뮬레이션을 통하여 동작특성을 보였다. 본 논문에서 제안한 곱셈기는 VCG의 기본 셀을 반복적으로 연결하여 구성하므로, 차수 m이 매우 큰 유한체상의 두 다항식의 곱셈에서 확장이 용이하며, VLSI에 적합하다.

• 한국컴퓨터정보학회논문지
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• 제20권2호
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• pp.63-70
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• 2015

최대 지배집합의 수인 도메틱 수 문제 (DNP)는 정확한 해를 다항시간으로 구하는 알고리즘이 존재하지 않아 NP-완전 문제로 알려져 있다. 본 논문은 DNP의 해를 다항시간으로 구하는 알고리즘을 제안하였다. 그래프의 최대 차수 ${\Delta}(G)$ 정점 $v_i$를 $D_i,i=1,2,{\cdots},k$의 지배집합의 원소로 선택하는 방법을 적용하고, $V_{i+1}=V_i{\backslash}D_i$의 축소된 그래프에 대해 $D_{i+1}$을 구하였다. 또한 $V{\backslash}D_i=N_G(D_i)$로 $D_i$가 지배집합으로 되는지 여부를 검증하였다. 제안된 알고리즘을 15개의 다양한 그래프에 적용한 결과 정확한 해를 다항시간 복잡도 O(kn)으로 구하는데 성공하였다. 결국, 제안된 알고리즘은 도메틱 수 문제가 P-문제임을 보였다.

• 대한수학회지
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• 제38권5호
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• pp.883-914
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• 2001

We present a survey of research results obtained for infra-nilmanifolds, their fundamental groups and some of their generalizations. This is presented from two different approaches and covers achievements obtained during the past four decades and showing a remarkable amount of mathematical interdisciplinarity. We go more in depth concerning the existence and construction of polynomial structures for these manifolds and groups, a direction where significant progress was made in the past few years. The bounded-degree polynomial structures developed by the authors triggered a number of challenging open problems. Also, their study already has lead to some interesting results concerning e.g. Anosov diffeomorphisms and expanding maps.

• 대한수학회논문집
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• 제33권2호
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• pp.639-650
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• 2018

Finding a solution for the Legendre equation is difficult. Especially if it is as a part of the Laplace equation solving in the electric fields. In this paper, first a problem of the generalized differential transform method (GDTM) is solved by the Sturm-Liouville equation, then the Legendre equation is solved by using it. To continue, the approximate solution is compared with the nth-degree Legendre polynomial for obtaining the inner and outer potential of a sphere. This approximate is more accurate than the previous solutions, and is closer to an ideal potential in the intervals.

• 대한수학회보
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• 제47권2호
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• pp.319-339
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• 2010

In this paper, we prove that if $f^nf'\;-\;P$ and $g^ng'\;-\;P$ share 0 CM, where f and g are two distinct transcendental meromorphic functions, $n\;{\geq}\;11$ is a positive integer, and P is a nonzero polynomial such that its degree ${\gamma}p\;{\leq}\;11$, then either $f\;=\;c_1e^{cQ}$ and $g\;=\;c_2e^{-cQ}$, where $c_1$, $c_2$ and c are three nonzero complex numbers satisfying $(c_1c_2)^{n+1}c^2\;=\;-1$, Q is a polynomial such that $Q\;=\;\int_o^z\;P(\eta)d{\eta}$, or f = tg for a complex number t such that $t^{n+1}\;=\;1$. The results in this paper improve those given by M. L. Fang and H. L. Qiu, C. C. Yang and X. H. Hua, and other authors.

• Kyungpook Mathematical Journal
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• 제50권1호
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• pp.1-5
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• 2010

Let P be a prime ideal of a commutative unital ring R; X an indeterminate; D := R/P; L the quotient field of D; F an algebraic closure of L; ${\alpha}$ ${\in}$ L[X] a monic irreducible polynomial; ${\xi}$ any root of in F; and Q = ${\alpha}$>, the upper to P with respect to ${\alpha}$. Then R[X]/Q is R-algebra isomorphic to $D[{\xi}]$; and is R-isomorphic to an overring of D if and only if deg(${\alpha}$) = 1.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제24권2호
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• pp.69-77
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• 2017

If q(z) is a polynomial of degree n with all zeros in the unit circle, then the self-reciprocal polynomial $q(z)+x^nq(1/z)$ has all its zeros on the unit circle. One might naturally ask: where are the zeros of $q(z)+x^nq(1/z)$ located if q(z) has different zero distribution from the unit circle? In this paper, we study this question when $q(z)=(z-1)^{n-k}(z-1-c_1){\cdots}(z-1-c_k)+(z+1)^{n-k}(z+1+c_1){\cdots}(z+1+c_k)$, where $c_j$ > 0 for each j, and q(z) is a 'zeros dragged' polynomial from $(z-1)^n+(z+1)^n$ whose all zeros lie on the imaginary axis.