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| Title | 熱帶亞純函數及其在差分方程之應用 Tropical Meromorphic Functions and Their Application on Difference Equations |
| Creator | 陳亮 Chen, Liang |
| Contributor | 蔡炎龍 Tsai, Yen Lung 陳亮 Chen, Liang |
| Key Words | 熱帶幾何 熱帶多項式 熱帶亞純函數 熱帶週期函數 tropical geometry tropical polynomial tropical meromorphic function tropical periodic function |
| Date | 2010 |
| Date Issued | 5-Oct-2011 16:18:29 (UTC+8) |
| Summary | 在這篇論文中，一個熱帶亞純函數(tropical meromorphic function)若給定有限個零根(roots)與極點(poles)還有它們的重數(multiplicities)，我們證明了這個熱帶亞純函數的存在與唯一性。另外，一個熱帶週期函數(tropical periodic function)若給定一個週期區間內的有限個零根與極點還有它們的重數，我們也找到了這個熱帶週期函數的一個簡單表達式。接著，給定一個一階線性差分方程(first-order linear difference equation)，我們討論了各種情況下的所有熱帶亞純函數解的表達式。最後，對於連續函數我們提供了一個它的熱帶近似函數，希望對於解差分方程的熱帶亞純函數解時能有所助益。 In this thesis, we find the formula of tropical meromorphic function by giving finite number of roots and poles (with multiplicities). We also find a simple formula for tropical periodic function by giving finite number of roots and poles (with multiplicities) during a period [0,T). We then discuss all cases of the tropical meromorphic solution functions of first-order linear difference equation. At last, we provide a tropical approximated function of a given continuous function. We hope it is helpful in solving the tropical meromorphic solution functions of a given difference equation. |
| 參考文獻 | Bibliography [1] Nathan Grigg. Factorization of tropical polynomials in one and several variables. Honor`s thesis,Brigham Young University, 2007. [2] Nathan Grigg and Nathan Manwaring. An elementary proof of the fundamental theorem of tropical algebra. arXiv:math.CO/0707.2591, 2008. [3] Rodney G. Halburd and Neil J. Southall. Tropical nevanlinna theory and ultradiscrete equations. Int. Math. Res. Not. IMRN, (5):887{911, 2009. [4] Ilpo Laine and Kazuya Tohge. Tropical nevanlinna theory and second main theorem. ArXiv e-prints, October 2009. [5] Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry. 2009. [6] Jurgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. ArXiv e-prints:math/0306366, 2003. [7] David Speyer and Bernd Sturmfels. Tropical mathematics. ArXiv e-prints:math/0408099, 2004. [8] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. ArXiv e-prints, January 2011. |
| Description | 碩士 國立政治大學 應用數學研究所 97751005 99 |
| 資料來源 | http://thesis.lib.nccu.edu.tw/record/#G0097751005 |
| Type | thesis |
| dc.contributor.advisor | 蔡炎龍 | zh_TW |
| dc.contributor.advisor | Tsai, Yen Lung | en_US |
| dc.contributor.author (Authors) | 陳亮 | zh_TW |
| dc.contributor.author (Authors) | Chen, Liang | en_US |
| dc.creator (作者) | 陳亮 | zh_TW |
| dc.creator (作者) | Chen, Liang | en_US |
| dc.date (日期) | 2010 | en_US |
| dc.date.accessioned | 5-Oct-2011 16:18:29 (UTC+8) | - |
| dc.date.available | 5-Oct-2011 16:18:29 (UTC+8) | - |
| dc.date.issued (上傳時間) | 5-Oct-2011 16:18:29 (UTC+8) | - |
| dc.identifier (Other Identifiers) | G0097751005 | en_US |
| dc.identifier.uri (URI) | http://nccur.lib.nccu.edu.tw/handle/140.119/51470 | - |
| dc.description (描述) | 碩士 | zh_TW |
| dc.description (描述) | 國立政治大學 | zh_TW |
| dc.description (描述) | 應用數學研究所 | zh_TW |
| dc.description (描述) | 97751005 | zh_TW |
| dc.description (描述) | 99 | zh_TW |
| dc.description.abstract (摘要) | 在這篇論文中，一個熱帶亞純函數(tropical meromorphic function)若給定有限個零根(roots)與極點(poles)還有它們的重數(multiplicities)，我們證明了這個熱帶亞純函數的存在與唯一性。另外，一個熱帶週期函數(tropical periodic function)若給定一個週期區間內的有限個零根與極點還有它們的重數，我們也找到了這個熱帶週期函數的一個簡單表達式。接著，給定一個一階線性差分方程(first-order linear difference equation)，我們討論了各種情況下的所有熱帶亞純函數解的表達式。最後，對於連續函數我們提供了一個它的熱帶近似函數，希望對於解差分方程的熱帶亞純函數解時能有所助益。 | zh_TW |
| dc.description.abstract (摘要) | In this thesis, we find the formula of tropical meromorphic function by giving finite number of roots and poles (with multiplicities). We also find a simple formula for tropical periodic function by giving finite number of roots and poles (with multiplicities) during a period [0,T). We then discuss all cases of the tropical meromorphic solution functions of first-order linear difference equation. At last, we provide a tropical approximated function of a given continuous function. We hope it is helpful in solving the tropical meromorphic solution functions of a given difference equation. | en_US |
| dc.description.tableofcontents | 1 Introduction 2 2 Background 4 2.1 Arithmetic . . . . . . . . . . . . . . . . . . . . 4 2.2 Some Equalities and Inequalities . . . . . . . . . 5 3 Tropical Meromorphic Functions 7 3.1 Tropical Polynomials in One Variable . . . . . . . 8 3.2 Tropical Meromorphic Functions . . . . . . . . . . 10 3.3 Extended Tropical Meromorphic Functions . . . . . 18 4 Tropical Periodic Functions 22 4.1 Generating Functions of Tropical Periodic Functions 23 4.2 R-linear Combination of Tropical 1-Periodic Functions 25 4.3 R-linear Combination of Tropical T-Periodic Functions 29 5 Application on Di erence Equations 35 5.1 Tropical Counterpart to The Exponential Function 36 5.2 Application on Di erence Equations: First Order 38 5.3 Tropical Approximated Function . . . . . . . . . . 42 | zh_TW |
| dc.language.iso | en_US | - |
| dc.source.uri (資料來源) | http://thesis.lib.nccu.edu.tw/record/#G0097751005 | en_US |
| dc.subject (關鍵詞) | 熱帶幾何 | zh_TW |
| dc.subject (關鍵詞) | 熱帶多項式 | zh_TW |
| dc.subject (關鍵詞) | 熱帶亞純函數 | zh_TW |
| dc.subject (關鍵詞) | 熱帶週期函數 | zh_TW |
| dc.subject (關鍵詞) | tropical geometry | en_US |
| dc.subject (關鍵詞) | tropical polynomial | en_US |
| dc.subject (關鍵詞) | tropical meromorphic function | en_US |
| dc.subject (關鍵詞) | tropical periodic function | en_US |
| dc.title (題名) | 熱帶亞純函數及其在差分方程之應用 | zh_TW |
| dc.title (題名) | Tropical Meromorphic Functions and Their Application on Difference Equations | en_US |
| dc.type (資料類型) | thesis | en |
| dc.relation.reference (參考文獻) | Bibliography | zh_TW |
| dc.relation.reference (參考文獻) | [1] Nathan Grigg. Factorization of tropical polynomials in one and several variables. Honor`s thesis,Brigham Young University, 2007. | zh_TW |
| dc.relation.reference (參考文獻) | [2] Nathan Grigg and Nathan Manwaring. An elementary proof of the fundamental theorem of tropical algebra. arXiv:math.CO/0707.2591, 2008. | zh_TW |
| dc.relation.reference (參考文獻) | [3] Rodney G. Halburd and Neil J. Southall. Tropical nevanlinna theory and ultradiscrete equations. Int. Math. Res. Not. IMRN, (5):887{911, 2009. | zh_TW |
| dc.relation.reference (參考文獻) | [4] Ilpo Laine and Kazuya Tohge. Tropical nevanlinna theory and second main theorem. ArXiv e-prints, October 2009. | zh_TW |
| dc.relation.reference (參考文獻) | [5] Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry. 2009. | zh_TW |
| dc.relation.reference (參考文獻) | [6] Jurgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. ArXiv e-prints:math/0306366, 2003. | zh_TW |
| dc.relation.reference (參考文獻) | [7] David Speyer and Bernd Sturmfels. Tropical mathematics. ArXiv e-prints:math/0408099, 2004. | zh_TW |
| dc.relation.reference (參考文獻) | [8] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. ArXiv e-prints, January 2011. | zh_TW |