(注:[HT08b] は、https://xorshammer.com/2008/08/23/set-theory-and-weather-prediction/ SET THEORY AND WEATHER PREDICTION XOR’S HAMMER Some things in mathematical logic that I find interesting WRITTEN BY MKOCONNOR Blog at WordPress.com. AUGUST 23, 2008 で 引用されており、かれの”Here’s a puzzle”の元ネタと思われる。
1) さて、まず、[HT08b] より (抜粋) P91 1. INTRODUCTION. We often model systems that change over time as functions from the real numbers R (or a subinterval of R) into some set S of states, and it is often our goal to predict the behavior of these systems. Generally, this requires rules governing their behavior, such as a set of differential equations or the assumption that the system (as a function) is analytic. 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, one could just define a function that diagonalizes against it: whatever the strategy predicts, define the function to be something else.” This argument, however, makes an appeal to induction: to diagonalize against the proposed strategy at a point t, we must have already defined our function for all s < t in order to determine what the strategy would predict at t.
In fact, the lack of well-orderedness in the reals can be exploited to produce a very counterintuitive result: there is a strategy for predicting the values of an arbitrary function, based on its previous values, that is almost always correct. Specifically, given the values of a function on an interval (?∞, t), the strategy produces a guess for the values of the function on [t,∞), and at all but countably many t, there is an ε > 0 such that the prediction is valid on [t, t + ε). Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strateg predicts some “ε-glimpse” of the future.
Nevertheless, we choose this presentation because we find it the most interesting, as well as pedagogically useful. For instance, “predicting the present” is a very natural way to think of the problem of guessing the value of f (t) based on f |(?∞, t).
2. THE μ-STRATEGY. (詳細は略すので、原文ご参照。ここに引用するには、数学記号が複雑過ぎるので。)
P92 3. PREDICTING THE PRESENT. (詳細は略すので、原文ご参照。ここに引用するには、数学記号が複雑過ぎるので。)
Corollary 3.4. If T = R and ∇ is <, then W0 is countable, has measure 0, and is nowhere dense.
What Corollary 3.4 tells us is that, if we model the universe as a function from the real numbers into some set of states, then the μ-strategy will correctly predict the present from the past on a set of full measure. (In the following section, we show that, on a set of full measure, it correctly predicts some of the future as well.) Note that these results concerning T = R are also valid when T is any interval of reals. One needs to be cautious about interpreting this as meaning that the μ-strategy is correct with probability 1. 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.
P95 W = {t ∈ R | the μ-strategy does not guess well at t }. Theorem 5.1. The set W is countable, has measure 0, and is nowhere dense.
(一部仮訳) 推論3.4が示していることは、実数からある状態の集合に対する関数とするuniverseをモデル化すると、μ戦略は過去からの現在を完全な尺度で正しく予測するということです。 (次のセクションでは、on a set of full measureで、将来の予測も正しく予測されることを示しています)。 T = Rに関するこれらの結果は、Tが実数の任意の区間である場合にも有効であることに留意されたい。
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),
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].
<結論> 1.以上より、[HT08b](XOR’S HAMMERのパズル元ネタ)は、著者自身の手([HT09]と[成書]と)で、否定されている。 ”The exact characterization of the error sets in this example (as scattered sets) was absent in [HT08b].” 2.元々、[HT08b]中で 「これをμ戦略が確率1で正しいと解釈することには注意が必要です。 固定されたfixed true シナリオの場合、区間[0,1](またはRにおいて、適切な確率分布の下で)において瞬間tをランダムに選択すると、 推論3.4は、μ戦略がtで確率1で正しいことを教えてくれる。 しかし、瞬間tを固定してランダムにfixed true シナリオを選択すると、そのシナリオの下でμ戦略が正しい確率は0であるか、または存在しないかもしれません ランダムなシナリオの概念をどのように定義するかによって異なります。」と注意を入れていて、完全に嵌まっている訳では無かったが しかし、その”1. INTRODUCTION”には、思わせぶりなことが書いてあり、ミスリードだろう。
3.XOR’S HAMMERは、勿論、きちんと[HT08b]中の注意書きは読んでいて、意識してあくまで、”Here’s a puzzle”と断っていることを注意しておく。
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
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)