現代数学の系譜工学物理雑談古典ガロア理論も読む46

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200: １３２人目の素数さん [sage] 2017/11/16(木) 11:57:32.61 ID:/MLxWF5k

>>179 追記

下記のピエロ紹介の論文（>>171）は、結構大学１年～２年の教育素材として、面白いと思う（＾＾

有理数の稠密性と無理数の関係
”DIOPHANTINE APPROXIMATION”
関数の連続・不連続
微分可能と不可能と
それに、関数ｙ＝1/x^ｖ の指数ｖによる属性の変化

さまざまな数学の要素が融合して
実に面白い素材だし

数学史の一コマにも使えるかな？（＾＾

http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION,
DIOPHANTINE APPROXIMATION,
AND A REFORMULATION
OF THE THUE-SIEGEL-ROTH THEOREM
JUAN LUIS VARONA
This paper has been published in Gazette of the Australian Mathematical Society, Vol-
ume 36, Number 5, November 2009, pp. 353{361.
Received 29 February 2008; accepted for publication 6 October 2009.

http://www.unirioja.es/cu/jvarona/
Juan L. Varona Dept. of Mathematics and Computation University of La Rioja

https://en.wikipedia.org/wiki/University_of_La_Rioja
University of La Rioja
(抜粋)
Type Public
Established 1992
Students 7,600
Address Avda. de la Paz, 93 26006, Logrono., Logrono, Spain
Website http://www.unirioja.es

The University of La Rioja (UR) is a public institution of higher education based in Logrono, La Rioja, Spain.
Inaugurated during 1992-1993 from various existing schools and colleges, it currently teaches Grades 19 adapted to the European Higher Education,
and a varied program of masters, summer courses and courses of Spanish language and culture for foreigners.
(引用終わり)

http://rio2016.5ch.net/test/read.cgi/math/1510442940/200

203: １３２人目の素数さん [sage] 2017/11/16(木) 13:53:40.78 ID:/MLxWF5k

>>200 補足
http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION,
DIOPHANTINE APPROXIMATION,
AND A REFORMULATION
OF THE THUE-SIEGEL-ROTH THEOREM 2009
抜粋引用
In the opinion of this author, fν is a very
interesting function, and it is worthwhile to continue analyzing its behaviour.
In this way, we find examples of functions whose properties about con-
tinuity and dierentiability are pathological at the same time. For every
ν > 0, the function fν is continuous at the irrationals and discontinuous
at the rationals. And, when ν > 2 (that is the most interesting case), we
prove that fν is dierentiable in a set Dν

It is astonishing that, dierentiability being a local concept, fν is dieren-
tiable almost everywhere in spite of the fact that it is not continuous at any
rational number.
We finish the paper by showing a reformulation of the Thue-Siegel-Roth
theorem in terms of the dierentiability of fν for ν > 2 (see Theorem 3
and the final Remark). It seems really surprising that a theorem about dio-
phantine approximation is equivalent to another theorem about the dier-
entiablity of a real function: a nice new connection between number theory
and analysis! As far as I know, this characterization of the Thue-Siegel-Roth
theorem has not been previously observed.

Remark 1. The pathological behavior of functions is a useful source of
examples that help to understand the rigorous definitions of the basic con-
cepts in mathematical analysis. In this respect, it is interesting to note that,
here, we have shown a kind of pathological behaviour that is dierent from
that of the more commonly studied: the existence of continuous nowhere
dierentiable real functions, whose most typical example is the Weierstrass
function

4. The theorem of Thue-Siegel-Roth revisited

http://rio2016.5ch.net/test/read.cgi/math/1510442940/203

514: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/11/23(木) 16:14:38.55 ID:A258vGqh

>>200 補足
>http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, JUAN LUIS VARONA 2009

References [4] A. Ya. Khinchin, Continued fractions, The University of Chicago Press, 1964. Reprint: Dover, 1997.

これの和訳がゲット！（＾＾
http://argent.shinshu-u.ac.jp/lecture/files/pdf/cfracb5.pdf
A. Ya. ヒンチン（Khinchin）著　連分数 （訳：乙部厳己）
（Khinchin, A. Ya., Continued fractions. With a preface by B. V. Gnedenko. Translated from the third (1961) Russian edition. Reprint of the 1964 translation. Dover Publications, Inc., Mineola, NY, 1997）

第3版への序
A. Ya. ヒンチン（Khinchin）による素晴らしい本のこの（第3）版は、著者の死後す
ぐにState Press for Physics and Mathematics によって引き受けられたものである。
このため、この本は私の頭文字（B.G.）の付けられた文献についての簡単な注意を除
けば何の変更もなされていない。

B. V. グネデンコ（Gnedenko）

Continued Fractions, Mineola, N.Y. : Dover Publications, 1997, ISBN 0-486-69630-8 (first published in Moscow, 1935)
(引用終り)

（参考）http://argent.shinshu-u.ac.jp/lecture/ 乙部厳己 Yoshiki OTOBE 信州大学理学部 数理・自然情報科学科
（乙部厳己）
ヒンチン（Aleksandr Yakovlevich Khinchin, 1894-1959）が亡くなって50年以上が経過しましたので、かつて訳したものを公開します。
注：かつて学部生の卒業研究の資料用に1週程度で訳したものですので、訳語・訳文の検討は一切なされておりません。また書き間違い等も残っています。その後一度大学院講義「力学系」として講義しましたので、もし要望があればそのときのメモを元に修正します。

つづく

http://rio2016.5ch.net/test/read.cgi/math/1510442940/514

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