臺灣博碩士論文加值系統

由球面的調和分析，我們希望能導出環面的調和分析，但是事實上非常困難，最後經由來理商 (Rayleigh quotient) 與找尋估計函數去估計精確的 "入" 的值。

The polar coordinate on the plane generalizes to spherical coordinate in 3-dimensional Euclidean space. However, it has many other generalizations other than the spherial. For instance, the cylindrical coordinate is the direct product of it with the z-axis. The name of such coordinate comes from its decomposition of the whole space into cylinders. Just like the spherical into spheres. It is known that the space can be also decomposed into tori of various sizes. So there is the toroidal coordinate corresponding to the tori. In this paper, we introduce the toroidal coordinate in space and attempt to find application in harmonic analysis on the torus.

In spherical coordinate, the harmonic homogenous polynomials restrict to spherical harmonics. The same theory doesn't apply successfully in our case. Because the radial doesn't decouple with the angular factors. Still,we can apply such coordinate to estimate the first nonzero eigenvalue of the Laplace
operator on the torus.

節次標題(Title) 頁次
誌謝(Acknowlegement)--------------------------------Ⅰ
摘要(Abstract)------------------------------ -------Ⅱ
目錄(Table of Content)------------------------------Ⅲ
第0章 引言(Introduction) ------------------------------------1
第1章 超越平面(Hyperbolic plane)-----------------------------1
第1.1節 定義(Definition) --------------------------------1
第1.2節 測地線方程式(Equation of geodesics) -------------2
第1.3節 在R3的旋轉(Rotation in R3) ----------------------2
第1.4節 在R3的conformal group----------------------------3
第2章 在R3上的環面座標(Toroidal coordinates of R3) ---------4
第2.1節 圓周的旋轉(Rotation of circles) ----------------4
第2.2節 附距與Laplacian (Metric and Laplacian) ---------4
第3章 環面(Torus) -------------------------------------------5
第3.1節 等溫線座標(Isothermal coordinates) -------------5
第3.2節 立體印射(Stereographic projrction) -------------5
第3.3節 Laplacian算子（Laplacian operator）-------------6
第3.4節 思通立維問題(Starm Liouville Problem) ----------7
第4章 估計(Estimation) -------------------------------------7
第4.1節 梯度(Gradient) ---------------------------------7
第4.2節 來理商(Rayleigh quotient) ----------------------8
第4.3節 估計函數(Testing function) ---------------------8
第4.4節 入１估計(Estimate of 入１) ---------------------8
第4.5節 結論(Conclusion) ------------------------------------9

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