| 姓名 |
黃聖傑(Huang Sheng-Jie)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
(Convergence rates of harmonic measures and extremal lengths of sets in the upper half plane)
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| 相關論文 | |
| 檔案 |
 [Endnote RIS 格式]
[Bibtex 格式]
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| 摘要(中) |
Beurling 估計表達出所有的調和測度都小於收斂速度的二分之一次方。
這篇論文中所做的是在上半平面線段調和測度收斂到零的速度跟收斂速度一致，以及所有上半平面的聯通子集收斂到零的速度有下界跟收斂半徑一致。
還有極值長度在上半平面的各種到零跟無限的收斂或發散情況探討。 |
| 摘要(英) |
Consider Beurling estimate we have w(z,B(zeta,r)cap partialOmega,Omega)
Now I will do harmonic measure convergence rate on upper-half plane with slash and extremal distance convergence or disconvergence rate on upper-half plane. |
| 關鍵字(中) |
★ 調和測度
★ 複分析
★ 極值長度 |
關鍵字(英) |
★ Harmonic measure
★ Extremal lengths
★ Complex analysis |
| 論文目次 |
中文摘要............................................................................................. i
英文摘要............................................................................................. iii
謝誌.................................................................................................... v
目錄.................................................................................................... vi
符號說明............................................................................................. ix
一、Preparation in Conformal Mapping .............................. 1
1.1 Schwarz-Christoel Mapping . . . . . . . . . . . . . 1
1.2 Riemann Mapping Theorem . . . . . . . . . . . . . 2
1.3 Caratheodory Convergence Theorem . . . . . . . . . 2
1.3.1 From fn ! f to Dn ! D . . . . . . . . . . . . . . . 2
1.3.3 From Dn ! D to fn ! f . . . . . . . . . . . . . . . 10
1.4 Prime End Theorem . . . . . . . . . . . . . . . . . 15
1.4.1 Pommerenke-Boundary Behaviour of Conformal Maps 15
二、Preparation in Harmonic Measure ................................ 29
2.1 The Half-Plane and the Disc . . . . . . . . . . . . . 29
2.2 Caratheodory Extension Theorem . . . . . . . . . . 35
2.3 Estimates for Harmonic Measure . . . . . . . . . . . 40
三、Preparation in Extermal Distance ................................ 47
3.1 De nitions and Examples . . . . . . . . . . . . . . . 47
3.2 Four Rules for Extremal Length . . . . . . . . . . . . 53
3.3 Extremal Metrics for Extremal Distance . . . . . . . 55
3.4 Extremal Distance and Harmonic Measure . . . . . . 57
四、Convergence Rate of Harmonic Measure ...................... 59
4.1 Relation between line on H map into D . . . . . . . . 59
4.2 For Vertical Line,Horizontal Line and H n D . . . . . 62
4.3 Lower Bound of Harmonic Measure for Slash . . . . . 64
4.4 Best Conclusion for Slash . . . . . . . . . . . . . . . 68
vii
4.4.1 Extend for conclusion of Slash . . . . . . . . . . . . 68
4.4.2 Convergence Rate for H n E is Simply Connected . . 70
五、Convergence Rate of Extremal Distance....................... 71
5.1 For R?? and [; + l] . . . . . . . . . . . . . . . . . . 71
5.1.1 For l = 1 and ! 0 . . . . . . . . . . . . . . . . . . 71
5.1.2 For = 1 and l ! 0 . . . . . . . . . . . . . . . . . . 73
參考文獻 . . . . . . . . . . . . . . . . . . 75 |
| 參考文獻 |
[1] J.B Garnett & D.E Marshall-Harmonic Measure, Cambridge University (2005)
[2] P.L.Duren-Univalent functions, Springer-Verlag (1983)
[3] J. Bak-Complex analysis(3rd), Springer (2010)
[4] W.Rudin-Real and complex analysis, McGraw-Hill Book company
[5] C.Pommerencke-Boundary Behaviour of Conformal Maps, Springer-Verlag
(1983)
[6] M.H.A. Newman-Elements of the topology of plane sets of points, Springer-
Verlag (1992)
[7] D. E. Marshall and S.Rohde-Convergence of a Variant of the Zipper Algorithm
for Conformal Mapping (2007)
[8] C. J. Bishop-Harmonic Measure: Algorithms and Applications (2018) |
| 指導教授 |
方向(Xiang Fang)
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審核日期 |
2018-7-20 |
| 推文 |
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