2.(文系) High level people たちの<数学ディベート>(もどき?)(>>8)について: >>492-494は、”uniform probability”を説明するための非数学的な例えの説明であって、そこに重箱の隅つつきの難癖をつけてもなんにもならんぜ 何も間違っていない。”uniform probability”の意味を理解していない、貴方たち(文系) High level peopleが、曲解して>>492-494のような難癖をつけているだけのことだ
1.[HT08b](XOR’S HAMMERのパズル元ネタ)は、XOR’S HAMMERのパズルそのものとは微妙に異なる 2.[HT08b]は、>>480 "if someone proposed a strategy for predicting the values of an arbitrary function based on its past values" とあるように、元々は、過去の関数値から、現在又は未来の関数値を予測するという話だった 3.但し、>>481 "For a fixed true scenario, if one randomly selects an instant t in the interval [0,1] (or in R, under a suitable probability distribution)"と一言注釈が入った 4.おそらく、XOR’S HAMMER氏は、ここをピックアップして、”XOR’S HAMMERの任意関数の数当て解法”パズル(>>56)を考案したんだろう が、当然(>>481) ”However, if one fixes the instant t, and randomly selects a true scenario, then the probability that the μ-strategy is correct at t under that scenario might be 0 or might not even exist, depending on how one defines the notion of a random scenario.” も読んでいて、あくまでパズルだと、”Here’s a puzzle:”(>>50より)を明記したわけだ
1.”>>479-485を、切り札にする”と言っても、言うほど簡単じゃない。 分量的にも大変だ。中途半端だと、議論の錯綜に輪を掛けることになる。 だから、PDFを3つ読み込まないといけなかった。 >>481の”However, if one fixes the instant t, and randomly selects a true scenario, ・・・ at t under that scenario might be 0 or might not even exist, depending on how one defines the notion of a random scenario.” には、早く気付いていたが、 他のPDFとの関連も確認する必要があった。
2.(文系) High level people たちの<数学ディベート>(もどき?)(>>8)は、全く面白くないんだよね。 自分達が、関連論文を読んで、紹介しようとしないから、話のレベルが全く上がらない。
> For a fixed true scenario, if one randomly selects an instant t in the interval [0,1] (or in R, under a suitable probability distribution), > then Corollary 3.4 does tell us that the μ-strategy will be correct at t with probability 1. > However, if one fixes the instant t, and randomly selects a true scenario, then the probability that the μ-strategy is correct at t under > that scenario might be 0 or might not even exist, depending on how one defines the notion of a random scenario.
スレ主の回答 「Aが直線であることを検証するためにはAの全ての点について直線の方程式を満たすか 確認しなければならない」 0511132人目の素数さん2017/11/23(木) 12:22:38.94ID:YPvALa6A ダメだ。。。スレ主のギャグには遠く及ばないorz 0512132人目の素数さん2017/11/23(木) 12:32:42.74ID:jgGp1UXf>>501-503 > For a fixed true scenario, if one randomly selects an instant t in the interval [0,1] (or in R, under a suitable probability distribution), > then Corollary 3.4 does tell us that the μ-strategy will be correct at t with probability 1. > However, if one fixes the instant t, and randomly selects a true scenario, then the probability that the μ-strategy is correct at t under > that scenario might be 0 or might not even exist, depending on how one defines the notion of a random scenario.
”集合 R^N からその元 s を一つ取り出すことを「s∈R^N を fix する」や「s∈R^N を固定する」などと言う”(下記前スレより)
1.∀s∈R^N or ∃s∈R^N どちらか? ということだね(^^ 2.”(1)FixされたR^Nに対して99/100が成り立つ からと言って (2)確率的に選ばれるR^Nに対して99/100が成り立つ は言えない、ということ。”(下記過去スレより)を説明する定義になっているのかな? 3.上記2の補足:”固定”とか”Fix”で、非可測集合が可測集合に変化すると言っているように見えるけど? どういうことなのかな? 無条件でそれが言えるなら、新説だろうね(^^ 4.”結局のところ、固定されたいかなるsでもν(s)≧99/100と言える”(下記過去スレより)って、”固定”の定義なしで数学の証明したんだね?(^^ 0513132人目の素数さん2017/11/23(木) 13:43:11.32ID:wNwKF+kQ ?u????????m????????????????????K?v???B ?@? @??????ixed true ?V?i???I?????A???0,1]?i??????? ??A?K???m???z????j?????u????????_????I????? A ?@?@???_3.4??A???????m????????????????B ?@?@????A????????????_????ixed true ?V?i???I? ?I?????A???V?i???I???????????m?????????A???? ?????????? ?@?@?????_????V?i???I??T?O????? ???`????????????B?v???????????A??S????????? ?????? ?@?@????A???h1. INTRODUCTION?h???A ?????????????????A?~?X???[?h??? 0514現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/23(木) 16:14:38.55ID: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)
https://en.wikipedia.org/wiki/Continued_fraction (抜粋) Contents [hide] 1 Motivation and notation 2 Basic formula 3 Calculating continued fraction representations 4 Notations for continued fractions 5 Finite continued fractions 6 Continued fractions of reciprocals 7 Infinite continued fractions and convergents 7.1 Properties 7.2 Some useful theorems 8 Semiconvergents 9 Best rational approximations 9.1 Best rational within an interval 9.2 Interval for a convergent 10 Comparison of continued fractions 11 Continued fraction expansions of π 12 Generalized continued fraction 13 Other continued fraction expansions 13.1 Periodic continued fractions 13.2 A property of the golden ratio φ 13.3 Regular patterns in continued fractions 13.4 Typical continued fractions 14 Applications 14.1 Square roots 14.2 Pell's equation 14.3 Dynamical systems 14.4 Eigenvalues and eigenvectors 15 Examples of rational and irrational numbers 16 History of continued fractions 17 See also 18 Notes 19 References 20 External links (引用終り)
http://www.maths.ed.ac.uk/~aar/papers/irwin.pdf Geometry of Continued Fractions MC IRWIN 著 The American Mathematical Monthly, Vol. 96, No. 8 , pp. 696-703 (Oct., 1989)
https://en.wikipedia.org/wiki/Nigel_Hitchin Nigel Hitchin (抜粋) Nigel James Hitchin FRS (born 2 August 1946) is the Savilian Professor of Geometry at Oxford University and a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.
Honours and awards In 2016 he received the Shaw Prize in Mathematical Sciences.[6] (引用終り)
(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x. (b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.” (引用終わり)
The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients (引用終わり)
On the complexity of algeraic numbers Iが(2007) On the complexity of algebraic numbers, IIが(2005)で、時間が逆転している
Annals of Mathの出版が遅いのか(^^ いや、ちょっと、連分数について調べているんだ(^^
http://adamczewski.perso.math.cnrs.fr/TMCF.pdf (avec Y. Bugeaud) Transcendence measure for continued fractions involving repetitive or symmetric patterns, J. Eur. Math. Soc. 12 (2010), 883--914.
http://adamczewski.perso.math.cnrs.fr/ Boris Adamczewski Institut Camille Jordan Universite Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 F-69622 Villeurbanne Cedex, France 0548現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/25(土) 21:10:43.16ID:QcNp0s4+>>545-546 おっちゃん、どうも、スレ主です。
(>>483より)<再録> (抜粋) 2) 次に[HT09] より (抜粋) P3126 The derivation of this from our main result uses the upward topology on α in which, as we mentioned, the scattered sets are the finite subsets of α. A known result that we extend here is Theorem 5.1 from [5] in which the present is predicted from an “infinitesimal” piece of the past, and the predictor is correct except on a countable set that is nowhere dense. In terms of our framework here, we have the topology on R in which the basic open sets are half-open intervals (w, x] (so f 〜x g if f and g agree on (w, x) for some w < x). It is known that the scattered sets here are countable and nowhere dense. The exact characterization of the error sets in this example (as scattered sets) was absent in [5].
[5] C. Hardin and A. Taylor, A peculiar connection between the axiom of choice and predicting the future, American Mathematical Monthly 115 (2008),
(>>484より)<再録> (抜粋) 3) 最後に[(成書)The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems]より (抜粋) P76 7.3 Corollaries
The second result we derive concerns the extent to which "the present can be predicted based on the past." Here, the exact characterization of the error sets occurs in Theorems 3.1 and 3.5 in [HT08b].
The derivation of this uses the topology on R in which the basic open sets are half-open intervals (w; x] (so f 〜x g if f and g agree on (w, x) for some w < x). It is known that the scattered sets here are countable and nowhere dense. The exact characterization of the error sets in this example (as scattered sets) was absent in [HT08b].
With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict. After all, if someone proposed a strategy for predicting the values of an arbitrary function based on its past values, a reasonable response might be, “That is impossible. Given any strategy for predicting the values of an arbitrary function, ・・・” (引用終り)
だから、 この大げさな、”INTRODUCTION”にある [HT08b] でのこの大げさに書いた ”what the strategy would predict at t.”の部分が すっかり成書では、削除された。
そして、成書には、”7 The Topological Setting ・・・71”として7章でTopologicalな条件付きの議論になったわけ。 つまり、”With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict.”ではなく、繰返すが、Topologicalな条件付きの議論に後退したわけだ(^^
fν(x) =0 if x ∈ R - Q(無理数) =1/q^ν if x = p/q ∈ Q, irreducible (有理数で既約分数) で
Theorem 1. For ν > 2, the function fν is discontinuous (and consequently not differentiable) at the rationals, and continuous at the irrationals. With respect the differentiability, we have: (a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x. (b) There exist infinitely many irrational numbers x such that fν is not differentiable at x. Moreover, the sets of numbers that fulfill (a) and (b) are both of them un-countable. (引用終り)
ここ、無理数を (a) For every irrational number x with bounded elementsと、 (b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.と 完全に2分したと読んだので、あとの測度論の下記Theorem 2
P6 Theorem 2. For ν > 2, let us denote Cν = {f ∈ R : fν is continuous at x } Dν = {f ∈ R : fν is differentiable at x }
Then, the Lebesgue measure of the sets R - Cν and R - Dν is 0, but the four sets Cν, R - Cν, Dν, and R - Dν are dense in R. (引用終り)
とが整合しないので、いろいろ調べていたんだ(>>556とか)(^^ ようやく分ったのは、 Dν≠{x |(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.} じゃないんだ!(^^
VARONA氏のP5 Lemma 3 g(t)について示しているように、”for almost all x”がDνなんだ。 つまり、”Dν={x | for almost all x at Lemma 3 }”みたい(^^
上記の”(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.”は、こんなのもあると、一例を示したと 1週間近く悩んでいたんだ(^^
R - Dν≠{x |(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x } でもないんだ・・、多分(^^ 0579132人目の素数さん2017/11/26(日) 19:06:38.74ID:eS22cW4G >つまり、”Dν={x | for almost all x at Lemma 3 }”みたい(^^ アホ丸出しw 0580現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/26(日) 19:08:40.25ID:1WQ1V5QH>>576 訂正
Dν≠{x |(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.} じゃないんだ!(^^ ↓ Dν≠{x |(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.} なんだ!(^^
R - Dν≠{x |(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x } でもないんだ・・、多分(^^ ↓ R - Dν≠{x |(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x } なんだ・・、多分(^^
原本PDFを見て貰った方が視認性は良いが、後の検索性のためにコピペする(^^ 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 2009 (抜粋) P7 4. The theorem of Thue-Siegel-Roth revisited
Or, equivalently, if x is an irrational algebraic number, there exists a positive constant C(x, α) such that |x - p/q |< C(x, α)/q^(2+α) (10) has no rational solution.
P8 Remark 3. We have proved Theorem 3 by using the Thue-Siegel-Roth theorem. But we have said that it is a reformulation. So, let us see how to deduce the Thue-Siegel-Roth theorem from Theorem 3. Given x algebraic and irrational, and ν > 2, Theorem 3 ensures that fν is differentiable at x, so there exists lim y→x {fν(y) - fν(x)}/(y - x) = f’ν (x). By approximating y → x by irrationals y, it follows that f’ν (x) = 0. Consequently, by approximating y → x by rationals, i.e., y = p/q, we also must have lim p/q→x {fν(p/q) - fν(x)}/(p/q - x ) = lim p/q→x (1/qν)/(p/q - x) = 0. Then, for every ε > 0, there exists δ > 0 such that 1/(q^ν) <= ε|p/q - x| when p/q ∈ (x - δ, x + δ). From here, it is easy to check that the same happens for every p/q ∈ Q, perhaps with a greather constant ε' in the place of ε. Thus, (10) with α = ν-2 and some positive constant C(x, α) = 1/ε' has no rational solution, and we have obtained the Thue-Siegel-Roth theorem. (引用終り)