請用此 Handle URI 來引用此文件：
http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/61827完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 蔡宜洵(I-Hsun Tsai) | |
| dc.contributor.author | Yue-Hong Li | en |
| dc.contributor.author | 李岳鴻 | zh_TW |
| dc.date.accessioned | 2021-06-16T13:14:46Z | - |
| dc.date.available | 2020-06-24 | |
| dc.date.copyright | 2020-06-24 | |
| dc.date.issued | 2020 | |
| dc.date.submitted | 2020-06-23 | |
| dc.identifier.citation | [1] S. MINAKSHISUNDARAM AND Å. PLEIJEL, Some Properties of The Eigenfunctions of The Laplace operator on Riemannian Manifolds, Canadian J.Math (1949), 242-256. [2] JOCHEN BRÜNING AND ERNST HEINTZE, The Asymptotic Expansion of MinakshisundaramPleijel in Tne Equivariant Case, Duke Mathematical Journal (1984) [3] JOCHEN BRÜNING, On The Asymptotic Expansion of Some Integrals, Arch. Math (1984)253-259 [4] PETER B. GILKEY, Invariance Theory, The Heat Equation. and The AtiyahSinger Index Theorem. (1996) [5] IVAN G.AVRAMIDI, Heat Kernel Asympcs on Symmetric Spaces.(2006) [6] HANS R.FISCHER, JERRY J.JUNGSTER, AND FLOYD L.WILLIAMS, The Heat Kernel on the Two Sphere, Journal of Mathematical analysis and applications (1985), 328-334 [7] MASAYOSHI NAGASE, Expressions of the Heat Kernels on spheres by elementary functions and their recurrence relations, Saitama Math.J. (2010) 25-34 [8] KEN. RICHARDSON, The Asymptotics of Heat Kernels on Riemannian foliations, GAFA, Geom. funct. anal. (1998), 356-401 [9] KEN. RICHARDSON, The Transverse geometry of Gmanifolds and Riemannian Foliations, Illinois J.Math(2001), 517-535 [10] KEN. RICHARDSON, Traces of Heat Operators on Riemannian Foliations, AMS(2009) [11] R. WONG, Asymptotic Approximations of integrals, SIAM(2001) | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/61827 | - |
| dc.description.abstract | 在這篇論文中，我們啟發自Jochen Brüning 及Ernst Heintze 的論文[2]，延伸他們對於式子1.1 的討論，研究在特定例子下該漸進展開的係數。我們透過拉普拉思算子、熱核與群作用的基本性質，將原式轉變成以幾何性質定義的量，並且不斷透過泰勒展開式將積分化簡以求得目標係數。最後我們會發現積分內的函數如何影響係數。 | zh_TW |
| dc.description.abstract | In this thesis, we are inspired by Jochen Brüning and Ernst Heintze’s work [2] and extend their result to achieve the coefficients of the asymptotic expansion of equation 1.1 in [2] in a specific condition. Our result will be based on their work. We will reduce the ordinary formula into an integration defined by some geometric objects via Laplacian, heat kernel and group action. Therefore, we use Taylor expansion to deal with the integration. We will find the relation between the functions in integration and the coefficients as our result. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T13:14:46Z (GMT). No. of bitstreams: 1 U0001-2106202014381900.pdf: 1386244 bytes, checksum: d7532ba59f9da609dc817ab0142a2925 (MD5) Previous issue date: 2020 | en |
| dc.description.tableofcontents | 口試委員會審定書. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 誌謝. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Heat Kernel on Riemannian Manifold and Sphere . . . . . . . . . . . 4 2.2 Group Action and Representation . . . . . . . . . . . . . . . . . . . 5 3 Computation for the coefficients . . . . . . . . . . . . . . . . . . . . . . 8 3.1 distance in the spherecoordinate . . . . . . . . . . . . . . . . . . . . 8 3.2 1variable integration . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.1 Study of 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.2 error function . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.3 Study of 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.4 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 2variables integration . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Reduction of the integration . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 | |
| dc.language.iso | en | |
| dc.subject | 群作用 | zh_TW |
| dc.subject | 熱方程式 | zh_TW |
| dc.subject | 漸進展開 | zh_TW |
| dc.subject | 球 | zh_TW |
| dc.subject | asymptotic expansion | en |
| dc.subject | group action | en |
| dc.subject | sphere | en |
| dc.subject | heat equation | en |
| dc.title | 球面熱核跡數在圓作用下之漸進展開 | zh_TW |
| dc.title | The Asymptotic Expansion of The Trace of Heat Kernel on S2 under S1-Action | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 108-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 王藹農(Ai-Nung Wang),鄭日新(Jih-Hsin Cheng) | |
| dc.subject.keyword | 漸進展開,熱方程式,球,群作用, | zh_TW |
| dc.subject.keyword | asymptotic expansion,heat equation,sphere,group action, | en |
| dc.relation.page | 26 | |
| dc.identifier.doi | 10.6342/NTU202001084 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2020-06-23 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
文件中的檔案：
| 檔案 | 大小 | 格式 | |
|---|---|---|---|
| U0001-2106202014381900.pdf 未授權公開取用 | 1.35 MB | Adobe PDF |
系統中的文件，除了特別指名其著作權條款之外，均受到著作權保護，並且保留所有的權利。