Linearly Independent Sets and Transcendental Numbers

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：71

、訪客IP：18.118.154.237

姓名

林宜萱(Yi-Hsuan Lin) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Linearly Independent Sets and Transcendental Numbers)

相關論文

★ 關於胡-黃-王猜測的研究	★ 關於 (2,n) 群試問題的研究
★ 關於方程式 x^2+11=p^n 的研究	★ 關於方程式2x^2+1=3^n的研究
★ On problems of certain arithmetic functions	★ 伽羅瓦理論
★ k階歐幾里得環	★ On a Paper of P. M. Cohn
★ On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic	★ On Generalized Euclidean Rings
★ ZCm 的理想環生成元個數之上限

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

摘要(中)

在線性代數中我們知道任何佈於一體 (over a field) 的向量空間 (vector space) 存在至少一組基底。這篇碩士論文的初始動機起源於一個由呂明光教授提出的問題：對佈於有理數體 Q 上的向量空間實數體 R，是否存在一組確切的基底？對此目前我們沒有給出一個答案，但在網路上搜尋到 F. G. Dorais 提供了一組確切佈於 Q 上的線性獨立實數子集 T，並且 |T| = |R|。從集合 T 我們衍生出一些例子，同樣是佈於 Q 上的線性獨立實數子集，並且其集合大小與 |R| 相同。由於代數數有無窮可數多個而實數有無窮不可數多個，另一個由呂明光教授提出的問題是決定 Dorais 提供的集合 T 中哪些數是超越數〔即，非代數數〕。為了回答這個問題，我們研讀 Edward B. Burger 和 Robert Tubbs 的書 Making transcendence transparent. An intuitive approach to classical transcendental number theory。這個問題還沒被解決，然而 Burger
與 Tubbs 的書中介紹了 Liouville 數〔一種特別的超越數〕，在此我們導出一些例子作為練習。

摘要(英)

It is known in linear algebra that every vector space over a field has a basis. The motivation of this thesis is to answer a question asked by Professor Ming-Guang Leu: Is there an explicit basis for the field R of real numbers over the field Q of rational numbers? To that we have yet no answer. However, it is found on the Internet that F. G. Dorais provides an explicit linearly independent subset T of R over Q with |T| = |R|. Inspired by the set T, we give some examples of linearly independent subsets of R over Q with the same cardinality as |R|. Since there are countably many algebraic numbers while there are uncountably many real numbers, another question asked by Professor Leu is to determine which number in the set T, given by Dorais, is a transcendental number (i.e., not an algebraic number). To answer the question, we study the book Making transcendence transparent. An intuitive approach to classical transcendental number theory by Edward B. Burger and Robert Tubbs. The question is not yet answered. However, in Burger and Tubbs′ book, Liouville numbers (a special type of transcendental numbers) are introduced, and we derive some examples of Liouville numbers as exercises.

關鍵字(中)

★ 線性獨立集
★ 超越數

關鍵字(英)

論文目次

Abstract (p.i)
Acknowledgements (p.iii)
Contents (p.iv)
Chapter 1: Introduction (p.1)
Section 1.1: Introduction (p.1)
Section 1.2: Preliminaries (p.2)
Chapter 2: Linearly independent subsets of R over Q (p.9)
Section 2.1: Key example (p.9)
Section 2.2: Examples inspired by Key example (p.11)
Section 2.3: More results (p.17)
Chapter 3: Transcendental numbers (p.20)
Section 3.1: Liouville′s Theorem with a proof (p.20)
Section 3.2: Examples of Liouville numbers (p.22)
Section 3.3: Roth′s Theorem (p.24)
Section 3.4: Mahler′s number M = 0.123456789101112... (p.25)
Section 3.5: Related notes (p.26)
References (p.30)

參考文獻

B. Adamczewski, On the expansion of some exponential periods in an integer base, Math. Ann. 346 (2010), 107-116.

É. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo 27 (1909), 247-271.

E. B. Burger and R. Tubbs, Making transcendence transparent. An intuitive approach to classical transcendental number theory, Springer-Verlag, New York, 2004.

J. W. S. Cassels, On a problem of Steinhaus about normal numbers, Colloq. Math. 7 (1959), 95-101.

D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, J. London Math. Soc. 8 (1933), 254-260.

A. H. Copeland and P. Erdös, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.

F. G. Dorais, Explicit big linearly independent sets, 2010, https://mathoverflow.net/questions/23202/explicit-big-linearly-independent-sets.

S. H. Friedberg, A. J. Insel and L. E. Spence, Linear algebra, 4th ed., Prentice Hall, Inc., Upper Saddle River, NJ, 2003.

J. B. Fraleigh, A First Course In Abstract Algebra, 7th ed., Addison-Wesley Publishing Co., 2003.

A. O. Gelfond, Transcendental and algebraic numbers, translated from the first Russian edition by L. F. Boron, Dover Publications, Inc., New York, 1960.

T. W. Hungerford, Algebra, reprint of the 1974 edition, Graduate Texts in Mathematics, 73, Springer-Verlag, New York-Heidelberg-Berlin, 1987.

N. Jacobson, Lectures in Abstract Algebra. Vol. II: Linear Algebra, reprint of the 1953 edition, Graduate Texts in Mathematics, 31, Springer-Verlag, New York-Heidelberg-Berlin, 1975.

M.-G. Leu, Lecture Notes, 2018.

M.-G. Leu, Lecture Notes, 2020.

J. Liouville, Sur des classes très-étendues de quantités dont la valeur n′est ni algébrique, ni même réductible à des irrationnelles algébriques, C. R. Acad. Sci. Paris 18 (1844), 883-885; 910-911.

K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Proc. Konink. Nederl. Akad. Wetensch., Amsterdam 40 (1937), 421-428.

D. A. Marcus, Number Fields, reprint of the 1977 edition, Universitext, Springer-Verlag, New York-Berlin-Heidelberg, 1987.

I. Niven, Irrational Numbers, reprint of the 1956 edition, The Mathematical Association of America, 1967.

K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20; 168.

W. Rudin, Principles of Mathematical Analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., Singapore, 1976.

W. M. Schmidt, On normal numbers, Pacific J. Math. 10 (1960), 661-672.

指導教授

呂明光(Ming-Guang Leu)

審核日期

2020-7-24

推文