• 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.

• ALGAE
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• 제37권2호
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• pp.175-183
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• 2022

Polar microorganisms produce physiologically active substances to adapt to harsh environments, and these substances can be used as biomedical compounds. The green microalga Micractinium variabile KSF0031, which was isolated from Antarctica, produced phytol, a natural antimicrobial agent. Furthermore, several polyunsaturated fatty acids (PUFAs), including omega-3, exhibit antioxidant properties. Here statistical methods (Plackett-Burman design and Box-Behnken design) were used to optimize the culture medium of KSF0031 to improve biomass production, and K₂HPO₄, MgSO₄·7H ₂O, and ammonium ferric citrate green (AFCg) were selected as significant components of the culture medium. Changes in the concentration of K₂HPO₄ and MgSO₄·7H ₂O as positive factors and AFCg as a negative factor affected cell growth to a remarkable degree. The biomass production in a 100 L culture using the optimized medium for 24 d at 18℃ was improved by 37.5% compared to that obtained using the original BG-11 medium. The quantities of PUFAs and phytol obtained were 13 mg g^-1 dry cell weight (DCW) and 10.98 mg g^-1 DCW, which represent improved yields of 11.70% and 48.78%, respectively. The results of this study could contribute to an improved production of phytol and fatty acids from Antarctic microalgae in the biomedical industry.

• Nonlinear Functional Analysis and Applications
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• 제27권1호
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• pp.141-148
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• 2022

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≤ 1, then Govil proved $$\max_{{\mid}z{\mid}=1}{\mid}p^{\prime}(z){\mid}{\leq}{\frac{n}{1+k^n}}\max_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$, provided |p'(z)| and |q'(z)| attain their maximal at the same point on the circle |z| = 1, where $$q(z)=z^n{\overline{p(\frac{1}{\overline{z}})}}$$. In this paper, we extend the above inequality to polar derivative of a polynomial. Further, we also prove an improved version of above inequality into polar derivative.

• Nonlinear Functional Analysis and Applications
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• 제28권3호
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• pp.731-751
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• 2023

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into L^r-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• Korean Journal of Mathematics
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• 제29권3호
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• pp.631-638
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• 2021

In this paper, some sharp inequalities for ordinary derivative P'(z) and polar derivative D_αP(z) = nP(z) + (α - z)P'(z) are obtained by including some of the coefficients and modulus of each individual zero of a polynomial P(z) of degree n not vanishing in the region |z| > k, k ≥ 1. Our results also improve the bounds of Turán's and Aziz's inequalities.

• 한국포장학회지
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• 제10권1호
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• pp.7-14
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• 2004

Migration behavior of selected solvents and food samples showed differences of the chemical structures and polarities, the food samples which have similar polar expresses more higher affinity than different polar degrees. Water which is polar has a highest partitioning coefficient values on polar isopropanol, and oil which is nonpolar has highest partitioning value on non-polar toluene. The increasing order of partitioning values was accord with increasing water contents in food samples. It is showed that the wheat flour with 13.2% moisture content has the highest partitioning coefficient values on the isopropanol with -OH. Kp value of sugar showed remarkable lower partitioning coefficient values than other food samples due to high degree crystallinity. This phenomenon can be predictable with ${\delta}$ values, because order of partitioning coefficient values which comes out through the experiment and the sequence of Hildebrand solubility parameter value difference between food sample and printing ink solvent correspond almost. This Hildebrand solubility parameter value can be easily applied to the food package industry because the effect of food-safety can be considered without passing through complicated steps by using this method.

• Nonlinear Functional Analysis and Applications
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• 제28권4호
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• pp.943-956
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• 2023

Let p(z) be a polynomial of degree n and for any complex number 𝛽, let D_𝛽p(z) = np(z) + (𝛽 - z)p'(z) denote the polar derivative of the polynomial with respect to 𝛽. In this paper, we consider the class of polynomial $$p(z)=(z-z_0)^s \left(a_0+\sum\limits_{{\nu}=0}^{n-s}a_{\nu}z^{\nu}\right)$$ of degree n having a zero of order s at z₀, |z₀| < 1 and the remaining n - s zeros are outside |z| < k, k ≥ 1 and establish upper bound estimates for the maximum of |D_𝛽p(z)| as well as |p(Rz) - p(rz)|, R ≥ r ≥ 1 on the unit disk.

• Ocean and Polar Research
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• 제30권4호
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• pp.407-418
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• 2008

Seawater samples were collected at discrete depths from five stations across the polar front in the Drake Passage (Antarctic Ocean) by the $20^{th}$ Korea Antarctic Research Program in December, 2006. Nitrate concentrations of seawater increase with depth within the photic zone above the depth of Upper Circumpolar Deep Water (UCDW). In contrast, ${\delta}^{15}N$ values of seawater nitrate decrease with depth, showing a mirror image to the nitrate variation. Such a distinct vertical variation is mainly attributed to the degree of nitrate assimilation by phytoplankton as well as organic matter degradation of sinking particles within the surface layer. The preferential $^{14}{NO_3}^-$ assimilation by the phytoplankton causes $^{15}{NO_3}^-$ concentration to become high in a closedsystem surface-water environment during the primary production, whereas more $^{14}{NO_3}^-$ is added to the seawater during the degradation of sinking organic particles. The water-mass mixing seems to play an important role in the alteration of ${\delta}^{15}N$ values in the deep layer below the UCDW. Across the polar front, nitrate concentrations of surface seawater decrease and corresponding ${\delta}^{15}N$ values increase northward, which is likely due to the degree of nitrate utilization during the primary production. Based on the Rayleigh model, the calculated ${\varepsilon}$ (isotope effect of nitrate uptake) values between 4.0%o and 5.8%o were validated by the previously reported data, although the preformed ${\delta}^{15}{{NO_3}^-}_{initial}$ value of UCDW is important in the calculation of ${\varepsilon}$ values.

• Nonlinear Functional Analysis and Applications
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• 제29권2호
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• pp.451-460
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• 2024

If a₀ + Σⁿ_ν=μ a_νz^ν, 1 ≤ µ ≤ n, is a polynomial of degree n having no zeroin |z| < k, k ≥ 1 and p'(z) its derivative, then Qazi [19] proved $$\max_{{\left|z\right|=1}}\left|p\prime(z)\right|\leq{n}\frac{1+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|k^{{\mu}+1}}{1+k^{{\mu}+1}+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|(k^{{\mu}+1}+k^{2{\mu}})}\max_{{\left|z\right|=1}}\left|p(z)\right|$$ In this paper, we not only obtain the L^r version of the polar derivative of the above inequality for r > 0, but also obtain an improved L^r extension in polar derivative.

• 터널과지하공간
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• 제20권4호
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• pp.241-251
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• 2010

극지는 학문적인 가치뿐만 아니라 무한한 자원의 공급처로서 각광받고 있는 추세이다. 극지와 극지 암석의 암석역학적 특성을 이해하기 위하여 남극대륙의 암석을 포함하여 극지 환경 조건에서 진행된 암석의 물성 변화에 관한 연구를 살펴보았다. 남극대륙은 바람이 매우 강하고 지구상에서 가장 기온이 낮으며 건조한 지역이다. 남극 현지 암석의 심도별 온도분포를 조사한 연구에서는 지표면과 가까운 얕은 심도에서는 동결-융해 과정이 반복되는 한편, 특정 깊이 이하에서는 항상 $0^{\circ}C$ 미만의 온도 분포를 나타내는 것으로 보고 하였다. 현지 암석의 온도분포와 대기 온도를 비교해 보았을 경우 상당한 차이를 나타내었다. 남극 사면의 기후에 의한 풍화정도를 알아보기 위하여 슈미트 반발경도와 taffoni의 분포를 조사한 연구가 있다. 그리고 실내 실험을 통해 암석 시료를 동결-융해 과정을 반복하여 풍화시킨 후 쇼어경도, 일축압축강도 등을 측정한 결과 동결-융해 횟수가 증가할수록 암석의 강도가 저하됨을 보고한 사례가 있다. 향후 극지암석에 대한 보다 상세한 암석역학적 물성 시험과 풍화에 관한 실내시험결과를 현장 암석의 물성변화에 적용시키는 연구가 필요할 것으로 판단된다.