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| 研究生： |
胡韋成 Hu, Wei-Cheng |
|---|---|
| 論文名稱： |
多維圓錐上的平均數不等式 Some Inequalities of Means on Circular Cones |
| 指導教授： |
陳界山
Chen, Jein-Shan |
| 口試委員： |
陳界山
Chen, Jein-Shan 張毓麟 Chang, Yu-Lin 杜威仕 Du, Wei-Shih |
| 口試日期： | 2024/06/27 |
| 學位類別： |
碩士 Master |
| 系所名稱： |
數學系 Department of Mathematics |
| 論文出版年： | 2024 |
| 畢業學年度： | 112 |
| 語文別： | 英文 |
| 論文頁數： | 28 |
| 中文關鍵詞： | 二階錐 、多維圓錐 、不等式 、調和平均數 、算術平均數 |
| 英文關鍵詞： | SOC, circular cone, harmonic means, arithmetic means, inequalities |
| DOI URL： | http://doi.org/10.6345/NTNU202400740 |
| 論文種類： | 學術論文 |
| 相關次數： | 點閱：48 下載：0 |
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這篇論文的主題涉及將二階錐(second order cone, SOC)上的相關定理延伸至多維圓錐(circular cone)上的類似定理。在二階錐 (SOC) 中，算術平均數、調和平均數與最大最小值有對應的大小關係，並可以形成一個不等式，本文中會將此不等式推廣至多維圓錐中，找出類似的大小關係，而在討論多維圓錐時，本文會將其分為兩個部分，即 𝜃 ∈ (0,π/4) 和 𝜃 ∈ (π/4,π/2)，進行各平均數不等式的討論，若與原不等式不相同則嘗試找出反例亦或是乘上特定的矩陣使其成立，並討論兩者之間的不同及相似之處。
The theme of this thesis involves extending some inequalities associated with second order cones (SOC) to the setting associated with circular cones.
Within SOC setting, there exists a relationship among the arithmetic mean, harmonic mean, maximum values and minimum values, which can be formulated into an inequality. This thesis aims to generalize this inequality to circular cones, identifying similar relationships. When discussing circular cones, the thesis will divide them into two parts: 𝜃 ∈ (0,π/4) and 𝜃 ∈ (π/4,π/2), for the discussion of various mean inequalities. If the results differ from the original inequalities, attempts will be made to find counterexamples or multiply with specific matrices to establish its validity, followed by discussions on the differences and similarities between the two cases.
J. S. Chen. SOC Functions and Their Applications. Springer, 2019.
J. S. Chen, H. F. Hung, and J. C. Zhou. Circular cone convexity and some inequalities
associated with circular cones. Journal of Inequalities and Applications, vol. 2013, Article ID 571, 17 pages, 2013.
J. S. Chen and J. C. Zhou. Properties of circular cone and spectral factorization associated with circular cone. J. Nonlinear Convex Anal 14 pp. 807-816, 2013.
J. S. Chen and J. S. Zhou. Monotonicity and circular cone monotonicity associated with circular cones. Set-Valued and Variational Analtsis, vol.25, no.2, pp. 211-232, 2017.
Z. Hao, C. T. Nguyen, and J. S. Chen. An approximate lower order penalty approach for solving second-order cone linear complementarity problems. Journal of Global Optimization, vol. 83, no. 4, pp. 671-697, 2022.
C. H. Huang, Y. L. Chang, and J. S. Chen. Some inequalities on weighted means and traces defined on second-order cone. Linear and Nonlinear Analysis, vol. 5, no. 2, pp. 221-236, 2019.
X. H. Miao, W. M. Hsu, C. T. Nguyen, and J. S. Chen. The solvabilities of three opti-mization problems associated with second-order cone. Journal of Nonlinear and Convex Analysis, vol. 22, no. 5, pp. 937-967, 2021.
X. H. Miao, K. Yao, C. Y. Yang, and J. S. Chen. Levenberg-marquardt method for absolute value equation associated with second-order cone. Numerical Algebra, Control and Optimization, vol. 12, no. 1, pp. 47-61, 2022.
