”有理数rが既約分数p/qで表されるとき、1/q^2”(>>146より)で、1/q^4くらいでどうかな? というのは、下記英文 Thomae's functionで、”f is not differentiable at all irrational numbers.”が参考になる Hurwitz's theoremから、(Thomae's function通り)1/qだと、”>=1/√5 *i”という評価になる で、1/q^nの指数nを大きくするというのは、ハイラー、ヴァンナーがヒントになる だが、1/q^2では足りないだろう
https://ja.wikipedia.org/wiki/%E3%83%88%E3%83%9E%E3%82%A8%E9%96%A2%E6%95%B0 トマエ関数 ↓ (英語版) https://en.wikipedia.org/wiki/Thomae%27s_function Thomae's function (抜粋) f is not differentiable at all irrational numbers. ・ According to Hurwitz's theorem, ・ Thus for all i,・・・>=1/√5 *i ≠ 0 and so f is not differentiable at all irrational x_0. (引用終わり) 0154132人目の素数さん2017/11/14(火) 16:11:08.07ID:jtNc+3xe>>153 関連 <追記> なお、上記のThomae's function引用の下記のURLが、ID登録を要求してくるので、フリーなサイトを探しておいた(^^ http://math.uga.edu/~pete/Kim99.pdf Kim, Sung Soo. "A Characterization of the Set of Points of Continuity of a Real Function." American Mathematical Monthly 106.3 (1999): 258-259. 0155132人目の素数さん2017/11/14(火) 16:11:22.11ID:jtNc+3xe>>152 (>>149補足) ヒント:同値類は、時間に依存しない(^^ 0156132人目の素数さん2017/11/14(火) 19:02:16.77ID:odeBuPNy>>155 時間の定義を述べよ 0157132人目の素数さん2017/11/14(火) 19:33:08.76ID:xUezoIEB>>156 定義の定義を述べよ 0158現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/14(火) 19:34:20.57ID:agSxZaXK>>156
問題に即して言えば (>>61より)"When, in step 3, Bob reveals {(x_0, f(x_0))|x_0≠x}," の前と後だが
This famous passage is the one where Galois proves the crucial lemma stating that any rational function of the roots can be expressed as a rational function of the Galois resolvent. Poisson (What about him?) had called Galois' prove insufficient. Galois, rather than elucidate his proof, laconically replied, "That remains to be seen. My opinion is in paragraph 37" (freely translated). It is easy to understand Poisson's position. Galois' proof can be regarded as as, at best, a sketch, and therefore is certainly "insufficient" if one is in any doubt as to the correctness of his theory and the accuracy of his reasoning. In his report to the Academy, Poisson said of Galois' memoir as a whole that << We have made every effort to understand Mr. Galois' proof. His argument are not clear enough, nor developed enough, for us to be able to judge their correctness... >>.
He hoped that Galois would improve and amplify his exposition of his work, but concluded "In the state in which it is now submitted to the Academy, we cannot recommend that you (Mr. Lacroix) give it your approval". At the time, confronted with an incomprehensible manuscript and a 19-year-old author who could well be asked to improve on it (and who was in trouble with the police to boot), one might well decide to recommend to one's colleagues that they not endorse it.
f is not differentiable at all irrational numbers.
All sequences of irrational numbers (ai≠ x0)_(i=1〜∞ ) converging to the irrational point x0 imply a constant sequence (f(ai)=0)_(i=1〜∞ ), identical to 0,
and so lim _(i→ ∞ )| (f(ai)-f(x0))/(ai-x0))|=0.
According to Hurwitz's theorem, there also exists a sequence of rational numbers (bi=ki/i)_(i=1〜∞ ),
converging to x0, with (ki∈ Z ,i∈ N )) (ki∈ Z ,i∈ N )) coprime and |ki/i-x0|< 1/(√ 5)* i^2).
Thus for all i,: |(f(bi)-f(x0))/ (bi-x0)| > (1/i - 0)/(1/((√ 5)* i^2)))= √ 5* i ≠ 0 and so f is not differentiable at all irrational x0. (引用終り) 0170現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/14(火) 22:14:42.31ID:agSxZaXK>>169 追加
Thomae's function なら、f(bi)=1/i だから、√ 5* i ≠ 0 ハイラー、ヴァンナーの「解析教程」下なら、f(bi)=1/i^2 だから、√ 5 ≠ 0
converging to x0, with (ki∈ Z ,i∈ N )) (ki∈ Z ,i∈ N )) ↓ converging to x0, with (ki∈ Z ,i∈ N ) 0174現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/15(水) 08:25:08.65ID:dypommzJ>>171 ピエロ、ありがとう たまらずPDFアップかな(^^
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.
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) 0201132人目の素数さん2017/11/16(木) 12:46:04.02ID:rWR6MHtP>>197 こいつは ぷ 0202132人目の素数さん2017/11/16(木) 13:26:41.61ID:/MLxWF5k 論争当事者同士だと、あれだけの発言で、だれか発言者が分かるのかね?(^^
私の立場だと、発言当事者も可能性としてはありうるからな〜(^^ 0203132人目の素数さん2017/11/16(木) 13:53:40.78ID:/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 0204132人目の素数さん2017/11/16(木) 14:32:12.79ID:j2ynrfH6>>201 むふ 0205132人目の素数さん2017/11/16(木) 17:15:31.23ID:/MLxWF5k>>204 やっぱり、「ぷふ」さんか。お元気そうでなによりです(^^
一つは、「Sergiu Hartの論文は論文誌に掲載されている 彼の著書でも紹介されている」(引用1)というウソ。彼は、典拠を示せない 一つは、「XOR’S HAMMERのHere’s a puzzle」が、Taylor氏の”A Study of Generalized Hat Problems ”にあるというウソ(引用2)
The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems (Developments in Mathematics) 2013 edition by Hardin, Christopher S., Taylor, Alan D. (2013) Hardcover Springer Verlag
1.(>>217に書いた)「同値類に時間依存性はない」って話も、理解できていなかったんだろうな(^^ 2.>>48にあるように、関数f 〜 g の同値類で、有限個の値のみ異なる同値類分類をすることを考える。
3.”in step 3, Bob reveals {(x0, f(x0)) | x0 ≠ x }, you know what equivalence class f is in, because you know its values at all but one point. ” 一つのやり方は、事前に全ての関数を類別して、代表を決めておくこと。これ、正攻法でパズルも同じ記載だ。 このやり方の問題点は、必要なのは、一つの関数fの同値類にすぎないのに、無駄な多数の同値類分類をすることだ。 もう一つのやり方は、事後的に”Bob reveals”の後に、問題の関数fの同値類のみを扱うこと。 こうすれば、無駄な作業はない。
4.さて、同値類分類の目的は、>>48にあるように、 ”Let g be the representative of that equivalence class that you picked ahead of time. Now, in step 4, guess that f(x) is equal to g(x).” とするための代表gを得ること。 しかも、代表gはなんでも可で、特別の制約なし。 さすれば、究極の手抜きは、”Bob reveals”の後に、fを得て、fの有限個の数値を適当に異なるようにして、チョコチョコとgを作る。 (関数fの同値類分類を完成させる必要さえない!!) このgを、さも事前に全ての関数の同値類を分類し、全ての代表を選んでおいた顔をして、「これが代表だよ、Bob!」と、gを出せば、Bobがびっくりするという仕掛けだ(^^