• Korean Journal of Mathematics
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• v.30 no.3
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• pp.503-512
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• 2022

In this paper, we are concerned with the problem of locating the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results generalize some recently proved results about the distribution of zeros of a quaternionic polynomial.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.04a
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• pp.41-45
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• 1997

To design a control system, it is a elementary point that the stability of the system should be guaranteed. Also, the phase of the system plays an important role for its frequence performance. In this paper, we present two stability criterion of repetitive control system with phase-lead and lag compensator. First, the stability criterion for the servo control system with phase-lead and lag compensator is shown by using small-gain theorem. Second, for the repetitive control system with the compensator, the stability criterion, also, is determined by using small-gain theorem. Two stability criterions show the same results that the stability depends on a coefficient of the phase-lead and lag compensator under some condition in servo control system and repetitive control system.

• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.691-700
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• 2022

Let p(z) be a polynomial of degree n having no zero inside the unit circle. Then for 0 < r ≤ 1, the well-known inequality due to Rivlin [Amer. Math. Monthly., 67 (1960) 251-253] is $$\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}{\geq}{\(\frac{r+1}{2}\)^n}\max\limits_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$. In this paper, we generalize as well as sharpen the above inequality. Also our results not only generalize, but also sharpen some known results proved recently.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.41-49
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• 2009

Let $\zeta$ be a fixed complex number. In this paper, we study the quantity $S(\zeta,\;n):=mas_{f{\in}{\Lambda}_n}\;|f(\zeta)|$, where ${\Lambda}_n$ is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given ${\zeta}\;{\in}\;{\mathbb{C}}$ and $n\;{\in}\;{\mathbb{N}}$, the quantity S($\zeta$, n) can be calculated. Then we compute the limit $lim_{n{\rightarrow}{\infty}}\;S(\zeta,\;n)/n$ for every ${\zeta}\;{\in}\;{\mathbb{C}}$ of modulus 1. It is equal to 1/$\pi$ if $\zeta$ is not a root of unity. If $\zeta\;=\;\exp(2{\pi}ik/d)$, where $d\;{\in}\;{\mathbb{N}}$ and k $\in$ [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin($\pi$/d)) and 1/(2d sin($\pi$/2d)) for d = 1, d even and d > 1 odd, respectively.