1.まず、そもそも話が有限ですむ場合は、”当たらない(=箱に数を入れる主題者勝率1、回答者勝率0)”ってことは、おっちゃん以外の全員が、同意している 実際にも、>>87に引用したSergiu Hart氏のPDF http://www.ma.huji.ac.il/hart/puzzle/choice.pdf? にも下記があるよ(これには全員同意だよ) P2 の最後 “Remark. When the number of boxes is finite Player 1 can guarantee a win with probability 1 in game1, and with probability 9/10 in game2, by choosing the xi independently and uniformly on [0, 1] and {0, 1, ・・・, 9}, respectively.”とある つまり、意訳すると “リマーク:箱の数が有限の場合、プレーヤー1は勝利を保証することができます。 [0、1]と{0、1、・・・、9}上で*)、xiを独立で一様に選択することによって、game1の勝利確率1とgame2の勝利確率9/10になる。”と 言い換えると、プレーヤー2の立場では、game1の勝利確率0とgame2の勝利確率1/10になる。 注*)、[0、1]はこの区間の任意の実数を、{0、1、・・・、9}は0〜9までの整数を、箱に入れるということ。 (引用終り)
で、Alan D. Taylor さんの2つの論文のPDFリンク切れているから、検索し直した 下記、ご参照
1) http://www.cs.umd.edu/~gasarch/ William Gasarch Professor of Computer Science Affiliate of Mathematics University of Maryland at College Park
http://www.cs.umd.edu/~gasarch/TOPICS/hats/hats.html Papers on Hat Problems I want to read by William Gasarch
21. An Introduction to Infinite Hat Problems by Christopher Hardin and Alan Taylor. HAT GAME- infinite number of people, need to get all but a finite number of them right. Needs AC. Infinite Hats and AC
http://www.cs.umd.edu/~gasarch/TOPICS/hats/infinite-hats-and-ac.pdf An Introduction to Infinite Hat Problems Chris Hardin and Alan Taylor THE MATHEMATICAL INTELLIGENCER 2008 Springer Science+Business Media, Inc
Alan Dana Taylor (born October 27, 1947) is an American mathematician who, with Steven Brams, solved the problem of envy-free cake-cutting for an arbitrary number of people with the Brams?Taylor procedure.
Taylor received his Ph.D. in 1975 from Dartmouth College.[2]
He currently is the Marie Louise Bailey professor of mathematics at Union College, in Schenectady, New York.
で、むしろ時枝記事に近いのは、君が>>295(>>304)で紹介した下記の方が、時枝に近いだろう ここでは、任意の関数f(x)の任意の貴方の選ぶ1点(”You pick an x ∈ R”)を、” whatever f Bob picked, you will win the game with probability 1!”、”it’s arbitrary: it doesn’t have to be continuous or anything”の条件で当てられるとあるよ
N⊂Rだから、”You pick an n ∈ N”とすれば、時枝記事の場合を含むことになろう で、時枝記事のように、どこの箱が当たるか分らず、また確率99/100に対して、これは自分で選んだxであり、”with probability 1!”だから、こちらの解法がよほど優れている
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 (抜粋) Here’s a puzzle: You and Bob are going to play a game which has the following steps.
1)Bob thinks of some function f: R → R (it’s arbitrary: it doesn’t have to be continuous or anything). 2)You pick an x ∈ R. 3)Bob reveals to you the table of values {(x0, f(x0))| x0 ≠ x } of his function on every input except the one you specified 4)You guess the value f(x) of Bob’s secret function on the number x that you picked in step 2.
You win if you guess right, you lose if you guess wrong. What’s the best strategy you have? This initially seems completely hopeless: the values of f on inputs x0 ≠ x have nothing to do with the value of f on input x, so how could you do any better then just making a wild guess? In fact, it turns out that if you, say, choose x in Step 2 with uniform probability from [ 0,1 ], the axiom of choice implies that you have a strategy such that, whatever f Bob picked, you will win the game with probability 1! つづく 0052現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/30(木) 22:28:15.89ID:IqNIthYM>>51 つづき
The strategy is as follows: Let 〜 be the equivalence relation on functions from R to R defined by f 〜 g iff for all but finitely many y, f(y) = g(y). Using the axiom of choice, pick a representative from each equivalence class.
In Step 2, choose x with uniform probability from [ 0,1 ]. When, 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. 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).
What is the probability of success of this strategy? Well, whatever f that Bob picks, the representative g of its equivalence class will differ from it in only finitely many places. You will win the game if, in Step 2, you pick any number besides one of those finitely many numbers. Thus, you win with probability 1 no matter what function Bob selects. (引用終り) 0053現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/30(木) 22:29:07.23ID:IqNIthYM>>52 つづき
先に私の見解を書いておくが、ピエロくんの紹介してくれた >>312 PDF が参考になるね(^^ The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems (Developments in Mathematics) 2013 edition by Hardin, Christopher S., Taylor, Alan D.
P9 ”In Chapter 7 we start to move further away from the hat problem metaphor and think instead of trying to predict a function's value at a point based on knowing (something about) its values on nearby points. The most natural setting for this is a topological space and if we wanted to only consider continuous colorings, then the limit operator would serve as a unique optimal predictor. But we want to consider arbitrary colorings. Thus we have each point in a topological space representing an agent and if f and g are two colorings, then f ≡a g if f and g agree on some deleted neighborhood of the point a. It turns out that an optimal predictor in this case is wrong only on a set that is "scattered" (a concept with origins going back to Cantor). Moreover, this predictor again turns out to be essentially unique, and this is the main result in Chapter 8.”
1)下記、XOR’S HAMMERのYou and Bobのpuzzleを、任意関数の数当て解法としよう。 記 (>>471より) 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 (抜粋) Here’s a puzzle: You and Bob are going to play a game which has the following steps.
2.任意関数の数当て解法は、射程として、可算無限個数列の数当て解法を含んでいるんだ。それを示そう 1)XOR’S HAMMERの任意関数の数当て解法は、”In Step 2, choose x with uniform probability from [ 0,1 ].”で、”Thus, you win with probability 1 no matter what function Bob selects.”なのだから 2)やり方は、>>483に書いたように、時枝の可算無限個との対応は、1/1,1/2,1/3,・・・1/n,・・・とすれば、全て[0,1]内の実数と対応がつく 3)数列 s = (s1,s2,s3 ,・・・,sn,・・・)から、 f(1)=s1,f(1/2)=s2,f(1/3)=s3 ,・・・,f(1/n)=sn,・・・となる関数f(x)を作れば良い。 関数はなんでも良いので、簡単に例えばf(1/2)とf(1/3)とを直線で結ぶ これで、時枝の可算無限個を、関数に埋め込めたので、XOR’S HAMMERの任意関数の数当て解法が適用できる 3)”you”は、好きな”1/n”を選べば、XOR’S HAMMERの任意関数の数当て解法で、当たる確率1だ
つづく
注)ここ、「“with uniform probability from [ 0,1 ].”を除いて、もとの問題設定通り、任意にxを選べるとすれば、」とするのが正確だったね。 “with uniform probability from [ 0,1 ].”だと、任意にxを選べないから。(^^ 0056現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/30(木) 22:32:09.79ID:IqNIthYM sage 0057現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/30(木) 22:32:29.83ID:IqNIthYM sage 0058現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/30(木) 22:32:35.74ID:IqNIthYM>>55 つづき
3.さて、XOR’S HAMMERの任意関数の数当て解法が、関数論の数理に反していることは明白だ ”Bob thinks of some function f: R → R (it’s arbitrary: it doesn’t have to be continuous or anything).”(>>471より) なのだから、解析関数でもなく、まして、連続でもない関数の値f(a)は、a以外の点の関数値が分かったところで、関数値f(a)は決まらない だから、XOR’S HAMMERの任意関数の数当て解法は、数理ではなくパズルであって、「選択公理と同値類を使えば、こんな奇妙は結論がもっともらしく見える」というところが面白いのだ
なぜなら、”XOR’S HAMMERの任意関数の数当て解法”は、たった1列で、かつ、決定番号を使わない! 一方、同値類 ”the equivalence relation on functions from R to R defined by f 〜 g iff for all but finitely many y, f(y) = g(y). ”と、当然選択公理も使うところが共通だから
(>>472より)”When, 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. ” なのだから(^^ 0064現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/30(木) 22:36:58.70ID:IqNIthYM>>63 関連
(>>472より)”When, 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. ” なのだから、x0を一つやれば、Bobのf(x)は、x0 以外全部分るんだ(^^
(>>471より)"In fact, it turns out that if you, say, choose x in Step 2 with uniform probability from [ 0,1 ]" だったでしょ?
簡単な話で、”choose x in Step 2 with uniform probability from [ 0,1 ]”だから、 Gameを、[ 0,1 ]の0から初めて1に達するまで、続ける x=0のときに、Bobのf(x)が分って、同値類が分って、代表f'(x)が決まる。あとを続ければ、Δf = f(x)−f'(x) は、”定義の通り” [ 0,1 ]では有限個しか不一致がないんだ
(>>667で、おれ) (抜粋) "In fact, it turns out that if you, say, choose x in Step 2 with uniform probability from [ 0,1 ]" は、飛ばして、「fと上記区間内の測度0の集合上のxで値が異なるだけのgを」に折り込んじゃったわけ?
実に、本質を捉えているので・・、 おれは賛成だけどね・・(^^ (引用終り)
(で、サイコパスのピエロ) >>671 名前:132人目の素数さん[] 投稿日:2017/11/10(金) 17:40:22.06 ID:lx5+65qp [8/9] >>667 >” choose x in Step 2 with uniform probability from [ 0,1 ]" は、飛ばして
参考文献 Andrews, George E. (1976), The Theory of Partitions, Cambridge University Press, ISBN 0-521-63766-X Andrews, George E.; Eriksson, Kimmo (2004), Integer Partitions (2nd ed.), Cambridge University Press, ISBN 0-521-60090-1 ジョージ・アンドリュース、キムモ・エリクソン 『整数の分割』 佐藤文広 訳、数学書房(出版) 白揚社(発売)、2006年5月。ISBN 978-4-8269-3103-8。 - 注記:原著第2版の翻訳。 (引用終わり) 0099現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/01(金) 10:56:32.81ID:Gaq1pHvm>>98 関連
https://en.wikipedia.org/wiki/Partition_(number_theory) Partition (number theory) (抜粋) Contents [hide] 1 Examples 2 Representations of partitions 2.1 Ferrers diagram 2.2 Young diagram 3 Partition function 3.1 Generating function 3.2 Congruences 3.3 Partition function formulas 3.3.1 Approximation formulas 3.3.2 Other recurrence relations 4 Restricted partitions 4.1 Conjugate and self-conjugate partitions 4.2 Odd parts and distinct parts 4.3 Restricted part size or number of parts 4.3.1 Asymptotics 4.4 Partitions in a rectangle and Gaussian binomial coefficients 5 Rank and Durfee square 6 Young's lattice 7 See also 8 Notes 9 References 10 External links
Notes 1 ^ Andrews 1976, p. 199.
References Andrews, George E. (1976). The Theory of Partitions. Cambridge University Press. ISBN 0-521-63766-X. Andrews, George E.; Eriksson, Kimmo (2004). Integer Partitions. Cambridge University Press. ISBN 0-521-60090-1. 0100現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/01(金) 11:14:42.07ID:Gaq1pHvm>>88 関連
補足 https://ja.wikipedia.org/wiki/%E7%A2%BA%E7%8E%87%E7%A9%BA%E9%96%93 確率空間 (抜粋) 定義 数学、特に確率論において、確率測度(かくりつそくど)とは、可測空間 (S, E) に対し、E 上で定義され P(S) = 1 を満たす測度 P のことである。 このとき、三つ組 (S, E, P) のことを確率空間と呼ぶ。さらに、集合 S を標本空間、S の元を標本あるいは標本点、完全加法族 E の元を事象あるいは確率事象とよぶ。また、E の元としての S を全事象という。 事象 E に対し、P の E における値 P(E) を、事象 E の起きる確率という。つまり、E は確率が定義できるものの集まりである。 必ずしも S の部分集合全てが事象とはならないことに注意されたい。 (引用終り)
厳密性を欠き、かつ間違っている(不正確)かも知れないが・・ あえて分かり易く書くと
1.Sを、全事象(”E の元としての S を全事象という”) 2.Eを、完全加法族で、Sの”可測”部分集合(但し、全事象Sをも含む)(”完全加法族 E の元を事象あるいは確率事象とよぶ”)*) 3.Pを、”確率”: P(E)(”事象 E に対し、P の E における値 P(E) を、事象 E の起きる確率という。つまり、E は確率が定義できるものの集まりである。”)
で、本題(>>135): 「お前は1回の試行ではuniform probabilityとは言えないと言ったのである choose x with uniform probability from [ 0,1 ] ならば[0 ,1]からuniform probabilityでxを選ぶという意味であり、 choose x with uniform probability from {1,2,3,4,5,6} ならば{1,2,3,4,5,6}からuniform probabilityでxを選ぶという意味である 試行の回数が1回ならばuniform probabilityではないというお前の主張は誤りである」
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
(参考)原文 [HT08b] Christopher S. Hardin and Alan D. Taylor. A peculiar connection between the axiom of choice and predicting the future. American Mathematical Monthly, 115(2):91{96, February 2008. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.365.7027&rep=rep1&type=pdf (抜粋) P93 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. (引用終り)
”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.”
上記>>145>>147-148>>154に書いた通りだよ 加えて、スレ46 https://rio2016.5ch.net/test/read.cgi/math/1510442940/483-484 「Taylor氏らは、[HT08b] の結論を否定している。([HT09] および(成書)The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems )」 つまりは、”Corollary 3.4 does tell us that the μ-strategy will be correct at t with probability 1.”(>>148)は、「数学的に無価値」でしたということですよ(^^
Research Articles My main research area is Banach space theory but, I have some work in real analysis and know some descriptive set theory as it applies to Banach space theory.
https://kbeanland.files.wordpress.com/2010/01/beanlandrobstevensonmonthly.pdf Modifications of Thomae’s function and differentiability, (with James Roberts and Craig Stevenson) Amer. Math. Monthly, 116 (2009), no. 6, 531-535. (抜粋) 3. A DENSE SET. While attempting to prove that T(1/n2) is differentiable on the irrationals, we discovered that quite the opposite is actually true. In fact, as the following proposition indicates, functions that are zero on the irrationals and positive on the rationals will always be non-differentiable on a rather large set.
Proposition 3.1. Let f be a function on R that is positive on the rationals and 0 on the irrationals. Then there is an uncountable dense set of irrationals on which f is not differentiable. (引用終り) 0246132人目の素数さん2017/12/02(土) 22:03:22.21ID:d9cBZA2m ID:Ph8fTUH9 は人格を認識する能力が不足しているようね 生きていく上で辛くない? 0247現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/02(土) 22:04:19.64ID:DyQaSaf9>>244 「ぷふ」さん、アシストありがとう!!(^^ 0248現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/02(土) 22:05:08.73ID:DyQaSaf9>>246 確かに、可哀想だよね(^^ 0249132人目の素数さん2017/12/02(土) 22:05:45.84ID:Ph8fTUH9 だから自演はやめろってw 自演中毒か?w 0250132人目の素数さん2017/12/02(土) 22:08:00.51ID:Ph8fTUH9 薬物と同じだなw 一回使ったが最後やめられなくなると言われるがまさにそれw 0251現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/02(土) 22:10:41.20ID:DyQaSaf9>>246 「ぷふ」さん、グッドタイミング過ぎるね(^^
まあ、そんなことをしなくても、P(x<y0)=y0/a で、a→∞とするとP(x<y0)→0 か・・(^^ 0276132人目の素数さん2017/12/03(日) 11:05:59.15ID:G2nPcR2G おっちゃんです。 他のスレで荒らしのような書き込みを見かけることがあるが、 もしかして、スレ主は自演して他のスレにも出没しているかい? 0277現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/03(日) 12:00:59.46ID:rPUpBQUT>>245 戻る (ピエロ) >さらにいえば、1/q^nを1/e^(-q)に置き換えても >リュービル数では微分不可能 https://kbeanland.files.wordpress.com/2010/01/beanlandrobstevensonmonthly.pdf (抜粋) Proposition 3.1. Let f be a function on R that is positive on the rationals and 0 on the irrationals. Then there is an uncountable dense set of irrationals on which f is not differentiable. Proof. Let (ri ) be an enumeration of the rationals. We recursively define a convergent sequence of rationals.
Proposition 4.2. 略
We finish by remarking on some obvious consequences of the previous propositions. First, for k <= 2, T(1/n^k ) is nowhere differentiable. By Roth’s Theorem, if α(an) > 2, T(ai ) is differentiable on the set of algebraic irrational numbers. T(1/n^9) is differentiable at all the algebraic irrationals, e, π, π^2, ln(2), and ζ(3), and not differentiable on the set of Liouville numbers. Finally, if α(ai ) = ∞, T(ai ) is differentiable on the set of all non-Liouville numbers. Since the set of Liouville numbers has measure zero, T(ai ) is differentiable almost everywhere. (引用終り)
ここ、Proposition 3.1. では、リュービル数は証明には使っていない。(”recursively define a convergent sequence of rationals”を使用) で、あとのProposition 4.2.の後で、Liouville numbersが、出てくるが、記載は上記の通り。
https://en.wikipedia.org/wiki/Liouville_number Liouville number (抜粋) In number theory, a Liouville number is an irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that
0<|x - p/q|< 1/q^n
A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time. (引用終り)
この定理を使うと、f:R → R であって、「xが有理数のとき不連続、xが無理数のとき微分可能」 となるものは存在しないことが即座に分かる。一応やってみると、そのような関数 f が存在したとすると、・・ (引用終り)
これ怪しいから、おれもstackexchangeのキーワードを使って、検索した。下記ヒットしたので貼る(^^ (抜粋) 1)”Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above. ** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)”とある 2)”[3] Tsuruichi Hayashi, "Eine stetige und nicht-differenzierbare function", Tohoku Mathematical Journal 1 (1911-12)”について、解説があったので、その部分を全文引用した 3)リプシッツ連続は、”** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]”とあるから、上記の定理と証明は怪しいかも(∵リプシッツ連続は微分可能と直結しないから)・・(^^ (参考 https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%97%E3%82%B7%E3%83%83%E3%83%84%E9%80%A3%E7%B6%9A リプシッツ連続
We would expect higher powers of f to be smoother, and this is what we find. Note that for each r > 0, the sets where f^r is continuous and discontinuous is the same as for f.
** For each 0 < r <= 2, f^r is nowhere differentiable.
** For each r > 2, f^r is differentiable on a set that has c many points in every interval.
** For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition. Heuer [15]
** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above.
Let w:Z+ --> Z+ be an increasing function that eventually majorizes every power function. Define f_w(x) = 0 for x irrational, f_w(0) = 1, and f_w(p/q) = 1/w(q) where p and q are relatively prime integers.
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)
Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.
THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
There are 22 items below. I found 4 of them on the internet, I provide the complete text for 9 of them, and I give some idea of what the remaining 9 items involve.
[3] Tsuruichi Hayashi, "Eine stetige und nicht-differenzierbare function", Tohoku Mathematical Journal 1 (1911-12), 140-142. [JFM 43.0482.03] [No submission date given.]
Hayashi mentions Lukacs' paper. I'm not sure if Hayashi is filling in some gaps from Lukacs' paper or extending the results in Lukacs' paper in some way. Hayashi's paper is in German, which I can't read. [Lukacs' paper is also in German, but in that case it was easy to figure out what Lukacs was doing. In this case, since Hayashi already knows of Lukacs' paper, the issue of what Hayashi is doing is not as immediately apparent to me.]
The complete text of the paper follows, with minor editing changes to accommodate ASCII format.
Im 70. Bande der Mathematische Annalen, S. 561, 1911, finden wir ein einfaches von Herrn Franz Lukacs gegebenes Beispiel einer Funktion, die in einer uberall dichten Menge unstetig und doch in einer anderen uberall dichten Menge differenzierbar ist. Nach der Lukacs-schen Methode, gebe ich im folgenden ein sehr einfaches Beispiel einer Funktion, die in einer uberall dichten Menge stetig uud nichtdifferenzierbar ist. Mein Beispiel wird als ein Resultat des Satzes von Liouville deduziert, wie Herr Lukacs's Beispiel.
Wer definieren die Funktion f(x) wie folgt: Fur jedes irrationale x sei f(x) = 0; wenn x rational und auf den kleinsten positiven Nenner gebracht = p/q ist, so sei f(x) = f(p/q) = 1/q.
Dann ist wie leicht ersichtlich, die so definierte Funktion fur jeden rationalen Wert von x and also in einer uberall dichten Menge unstetig, und doch fur jeden irrationalen Wert von x stetig. Die Funktion f(x) ist fur jeden nicht-algebraischen, i.e. transzendentalen Wert von x, der ein Element der von Liouville angegebenen Menge ist, und also in einer uberall dichten Menge, (1) nicht-differenzierbar.
(1) Vgl. A. Schonflies: Die Entwickelung der Lehre von Punktmannigfaltigkeiten, Jahresbaricht der Deutschen Mathematiker-Vereinigung. 8ter Band, S. 103, 1900.
Wenn p/q - b < 0, i.e. als den ruckwarts genommenen Differenzenquotient betrachtet, ist
H(p/q, b) = (1/q) / (p/q - b) < 0
und ist
p/q - b > -1 / (Mq^n).
Also ist
H(p/q, b) < (1/q) / (-1/Mq^n) = -Mq^(n-1).
Daher ist der ruckwarts genommene Differentialquotient negativ und wird unendlich.
Die Funktion ist fur alle Argumente nicht-differenzierbar, nicht nur fur transzendente Zahlen. Dar Beweis ist sehr einfach folgender.
Sie b ein irrationaler Wert und x ein irrationaler Nachbarwert, dann ist f(x) - f(b) = 0 und daher der Differenzenquotient = 0. Andererseits lasst sich x durch eine Reihe rationaler Zahlen, die Naherungsbruche des Kettenbruchs fur b, in der weise annahern, dass, wenn p_n/q_n ein solcher Naherungsbruch in reduzierter Form ist, die Ungleichung besteht
| b - p_n/q_n | < 1 / (q_n)^2.
Daher wachst der Differenzenquotient
[ f(p_n/q_n) - f(b) ] / [ p_n/q_n - b ]
uber alle Grenzen mit wachsendem n. Es kann daher kein Differentialquotient existieren.
Proposition 3.1. Let f be a function on R that is positive on the rationals and 0 on the irrationals. Then there is an uncountable dense set of irrationals on which f is not differentiable.
The ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and f(x) = 1/q^r if x = p/q where p and q are relatively prime integers with q > 0.
で、指数rで、関数の特性が類別されているだろ(下記) で、(抜粋) 1)** For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition. Heuer [15]
2)** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
3)** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
つまり、0 < r < 2でLipschitzでなく、r = 2でLipschitz、r > 2でdifferentiableだと。 だから、指数r依存性があるよと。指数r依存性とは、如何に早く0(ゼロ)に減衰するかだ
そして、>>282のピエロの証明は、 4)Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above. つまり、1/q^rという関数(多項式の逆数)よりも、早く減衰するときにも、not differentiable な無理数が残るという場合の証明だ
(抜粋) ”** For each 0 < r <= 2, f^r is nowhere differentiable.
** For each r > 2, f^r is differentiable on a set that has c many points in every interval.
The results above can be further refined.
** For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition. Heuer [15]
** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above.
Let w:Z+ --> Z+ be an increasing function that eventually majorizes every power function. Define f_w(x) = 0 for x irrational, f_w(0) = 1, and f_w(p/q) = 1/w(q) where p and q are relatively prime integers.” (引用終り)
http://www-03.ibm.com/press/jp/ja/pressrelease/52794.wss IBM Z (z14) プレスリリース IBM 2017年7月18日 (抜粋) 妥協なきセキュリティー z14は初めて、システムに関わる全てのデータをOSレベルでハードウェア暗号化の機能を使用して一度に暗号化できるようになりました。 現行の暗号化ソリューションはCPU負荷が高くシステムの処理性能や応答時間に影響を与え、また暗号化するフィールドの選定や管理に多くの工数がかかっていました。 今回、暗号化アルゴリズム専用の回路を4倍にすることで暗号化処理性能を前モデルであるIBM z13比で最大7倍に増強した結果、クラウド規模のバルク暗号化が可能となりました。
機械学習による新たな価値の創造 z14は前モデルの z13比で約3倍となる32TBのメモリーが最大で搭載可能となり、分析処理の応答時間の短縮およびスループットの増大を実現しています。 またzHyperLinkを利用することでストレージ・エリア・ネットワーク応答時間をz13に比べて10分の1に短縮し、アプリケーションの応答時間を半減します[4]。 これらのマシン性能向上に加えて、本年2月に発表されたIBM Machine Learning for z/OSを用いた機械学習により、業務分析モデルの作成、学習、展開を自動化することが可能になり、リアルタイム分析の効率性が大幅に向上します。
ペレルマン論文に対する検証が複数の数学者チームによって試みられた。原論文が理論的に難解でありかつ細部を省略していたため検証作業は難航したが、2006年5?7月にかけて3つの数学者チームによる報告論文が出揃った。 ・ブルース・クライナーとジョン・ロット, Notes on Perelman's Papers(2006年5月) ペレルマンによる幾何化予想についての証明の細部を解明・補足 ・朱熹平と曹懐東、A Complete Proof of the Poincare and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow(2006年7月、改訂版2006年12月) ペレルマン論文で省略されている細部の解明・補足 ・ジョン・モーガンと田剛、Ricci Flow and the Poincare Conjecture(2006年7月) ペレルマン論文をポアンカレ予想に関わる部分のみに絞って詳細に解明・補足 これらのチームはどれもペレルマン論文は基本的に正しく致命的誤りはなかったこと、また細部のギャップについてもペレルマンの手法によって修正可能であったという結論で一致した。これらのことから、現在では少なくともポアンカレ予想についてはペレルマンにより解決されたと考えられている。 (引用終り) 0343132人目の素数さん2017/12/05(火) 07:44:07.76ID:/o47Z1m6>>1 数学の知識で人を殴るのは道を尋ねられたらムカつくから殴るのと同じくらいの幼稚な行動なのでは
理解できないから、またまた検索したら、下記ヒットした。ご存知と思うが、自分のメモとして貼る 後述”It is a hard (and often open) problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers.”とか
https://en.wikipedia.org/wiki/Lottery Lottery (抜粋) Contents [hide] 1 Classical history 2 Medieval history 3 Ticket gallery 4 Early modern history 4.1 France, 1539?1789 4.2 England, 1566?1826 4.3 Early United States 1612?1900 4.4 German-speaking countries 4.5 Spain, 1763 5 Modern history by country 5.1 Australia 5.2 Canada 5.3 Mexico 5.4 Spain 5.5 Thailand 5.6 United Kingdom 5.7 United States 6 Mathematical analysis 6.1 Probability of winning 7 Scams and frauds 8 Payment of prizes 9 See also 10 References 11 Further reading 12 External links (引用終り)
https://en.wikipedia.org/wiki/Lottery (抜粋) Early modern history France, 1539?1789 King Francis I of France discovered the lotteries during his campaigns in Italy and decided to organize such a lottery in his kingdom to help the state finances. The first French lottery, the Loterie Royale, was held in 1539 and was authorized with the edict of Chateaurenard. This attempt was a fiasco, since the tickets were very costly and the social classes which could afford them opposed the project. During the two following centuries lotteries in France were forbidden or, in some cases, tolerated.
Mathematical analysis The purchase of lottery tickets cannot be accounted for by decision models based on expected value maximization. The reason is that lottery tickets cost more than the expected gain, as shown by lottery mathematics, so someone maximizing expected value should not buy lottery tickets. Yet, lottery purchases can be explained by decision models based on expected utility maximization, as the curvature of the utility function can be adjusted to capture risk-seeking behavior. More general models based on utility functions defined on things other than the lottery outcomes can also account for lottery purchase. In addition to the lottery prizes, the ticket may enable some purchasers to experience a thrill and to indulge in a fantasy of becoming wealthy. If the entertainment value (or other non-monetary value) obtained by playing is high enough for a given individual, then the purchase of a lottery ticket could represent a gain in overall utility. In such a case, the disutility of a monetary loss could be outweighed by the combined expected utility of monetary and non-monetary gain, thus making the purchase a rational decision for that individual. (引用終り)
https://en.wikipedia.org/wiki/Lottery_mathematics Lottery mathematics (抜粋) Contents [hide] 1 Choosing 6 from 49 2 Odds of getting other possibilities in choosing 6 from 49 3 Pick8-32 odds and calculations 4 Powerballs And Bonus Balls 5 Minimum number of tickets for a match 6 References 7 External links
Minimum number of tickets for a match It is a hard (and often open) problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers. In the 5-from-90 lotto, the minimum number of tickets that can guarantee a ticket with at least 2 matches is 100.[3]
References 1. Zabrocki, Mike (2003-03-01). "Calculating the Probabilities of Winning Lotto 6/49,Version 3" (PDF). Retrieved 2016-08-14. http://garsia.math.yorku.ca/~zabrocki/math5020f03/lot649/lot649v3.pdf
http://garsia.math.yorku.ca/~zabrocki/math5020f03/ Math 5020 Fundamentals of Mathematics for Teachers Professor Mike Zabrocki
(March 1, 2004) I revised the draft of the explanation of Lottery 6/49 to produce version 3. At this point I don't have much momentum on this project, but please offer your comments on the forum (I got none this last week except for one negative one). See my remarks on the forum. I will bring this up in class tonight.
https://en.wikipedia.org/wiki/Fierz_identity Fierz identity (抜粋) In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz.
There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1 dimensions.
Spinor bilinears can be thought of as elements of a Clifford Algebra. Then the Fierz identity is the concrete realization of the relation to the exterior algebra. The identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according to the following table. (引用終り)
これ、4番目(PDFが直接落ちるのでURL取れず。下記で代用) a mathematical model for the lottery - RACO www.raco.cat/index.php/Questiio/article/download/26594/26428 このページを訳す MS Nikulin 著 - ?1992 - ?関連記事 the prize winners, being those holding the tickets with those corres- ponding numbers.” From “Educated Guessing” Samuel Kotz (1983, Marcel Dekker):. “A lottery is a game of chance with low stakes and potentially high winnings, which account for the widespread appeal of this type of gambling. In its simplests form, a player bets on a number and wins if the state also selects that number. While we usually view a lottery as a game, many applications exist in the real world. For example,.
これ、5番目(PDFではないが、Lottery Software が詳しそうだ) http://saliu.com/strategy.html Basics of a Lotto Strategy Based on: Sums (Sum-Totals); Odd Even; Low High Numbers By Ion Saliu, Founder of Lottery Mathematics (抜粋) I. Introduction to Lottery Strategies, Filtering, Number Grouping II. Pick-3 Lottery Software for Low or High, Odd or Even Digit Grouping III. Lotto Software for Low / High, Odd / Even Numbers, Plus Lotto Skipping IV. True Lottery Filters, Filtering to Create the Best Lotto, Lottery Strategies, Systems V. Essential Resources in Lotto, Lottery Strategy, Systems, Software
Resources, links to the best in lotto software, lottery strategies.
つづく 0420現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/07(木) 22:23:23.82ID:jVxSMgKf>>419 つづき 5. Resources in Lottery Software, Systems, Strategies, Lotto Wheeling Introduction to Lottery Mathematics: Probabilities, Appearance, Repeat, Affinity or Number Affiliation, Wheels, Systems, Strategies. The Starting Strategy Page: Lottery Software, Strategy, Systems. Presenting software to create free winning lotto, lottery strategies, systems based on mathematics. Get your lotto systems or wheels, the best lottery, lotto software, combinations, winning numbers. Lotto, Lottery Software, Excel Spreadsheets: Programming, Strategies. Read a genuine analysis of Excel spreadsheets applied to lottery and lotto developing of software, systems, and strategies. Combining Excel analysis with powerful lottery and lotto software programmed by this author, Parpaluck. MDIEditor Lotto WE: Lottery Software Manual, Book, ebook, Help. ~ Also applicable to LotWon lottery, lotto software; plus Powerball, Mega Millions, Euromillions. Visual Tutorial, Book, Manual: Lottery Software, Lotto Apps, Programs. Basic Manual for Lotto Software, Lottery Software. Sum-Totals for Lottery, Lotto Games ? Pick 3 4 Lotteries, Lotto 5, 6, Powerball, Mega Millions, Euromillions. Lotto Software for Groups of Numbers: Odd, Even, Low, High, Sums, Frequencies, User's Groups. Lottery Software Sum-Totals, Sums: Lotto, Powerball, Mega Millions, Euromillions. Lottery Skip Systems: Lotto, Powerball, Mega Millions, Euromillions. Lotto, Lottery Strategy in Reverse: Not-to-Win Leads to Not-to-Lose or WIN. Lottery Utility Software: Pick-3, 4 Lottery, Lotto-5, 6, Powerball, Mega Millions, Euromillions. Theory, Analysis of Deltas in Lotto, Lottery Software, Strategy, Systems. The Best Strategy for Lottery, Gambling, Sports Betting, Horse Racing, Blackjack, Roulette. Lotto Software for Groups of Numbers. "The Start Is the Hardest Part":Play a Lotto Strategy, Lottery Strategies Download lottery software, lotto software つづく 0421現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/07(木) 22:27:03.74ID:jVxSMgKf sage 0422現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/07(木) 22:28:11.68ID:jVxSMgKf>>420 つづき (上記の原文にはURLのリンクが貼ってあるよ)
あと、数学公式としては ”余録3: http://mathworld.wolfram.com/HypergeometricDistribution.html Hypergeometric Distribution -- from Wolfram MathWorld (抜粋) It therefore also describes the probability of obtaining exactly i correct balls in a pick-N lottery from a reservoir of r balls” で尽きているような気がしてきたね 0429現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/08(金) 07:31:01.17ID:pI4TAlAF>>428 補足
>当然、内容は数分間の流し読みですよ(^^
キーワード下記 PDF Lottery mathematics Odds of getting other possibilities summation 78 件 (0.55 秒)
<前振りで数学的な構造> (>>284-285より) http://mathforum.org/kb/message.jspa?messageID=5432910 Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) The ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and f(x) = 1/q if x = p/q where p and q are relatively prime integers with q > 0.
** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above.
Let w:Z+ --> Z+ be an increasing function that eventually majorizes every power function. Define f_w(x) = 0 for x irrational, f_w(0) = 1, and f_w(p/q) = 1/w(q) where p and q are relatively prime integers.
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25) (引用終り)
趣旨を日本語にすると ruler functionとか、改良トマエ関数で、 f(x) = 1/q if x = p/q ↓ f^r = 1/q^r となって
1)指数r=2なら:nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. 2)指数r > 2なら:differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. 3)指数1/q^rより早く減衰する関数1/w(q) :differentiable on a set whose complement has Hausdorff dimension zero. (前振り終り)
で、「証明すべきこと」は、1/q^rで、Hausdorff dimension 1 - 2/rで、rが大きくなると、どんどんHausdorff dimensionが1に近づく。つまり、differentiableな範囲が大きくなる 指数1/q^rより早く減衰する関数1/w(q)では、”a set whose complement has Hausdorff dimension zero”ですよ
しかし、指数1/q^rより早く減衰する関数1/w(q)でも、微分不可の部分が残って、Hausdorff dimension zeroにもかかわらず、 ”Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.”(>>285より) だと。つまり、証明すべきは、ここで、”指数1/q^rより早く減衰する関数1/w(q)でも、微分不可の部分が残って、Hausdorff dimension zeroにもかかわらず、「fail to be differentiable is large」なのだ”ということなのだ
重複を厭わず、下記追加引用 (下記より) ”THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.”
ここで、”Let g be continuous and discontinuous on sets of points that are each dense in the reals.” とあるでしょ。この”each dense in the reals”を覚えておいてね。あとで使う(^^
(>>285より) http://mathforum.org/kb/message.jspa?messageID=5432910 Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) ** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)
Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.
THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
There are 22 items below. I found 4 of them on the internet, I provide the complete text for 9 of them, and I give some idea of what the remaining 9 items involve.
1.で、(>>443)英文では”each dense in the reals” ”THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.”
という例の場合は、R − ∪[p∈Q] { p } には開区間が全く存在しない。 だから、そのイメージの仕方は間違っている。 0465現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 22:18:51.30ID:OrUOLzdR 哀れな素人さんのために Philosophy本だが、検索ヒットしたので貼る(^^ http://publish.uwo.ca/~jbell/The%20Continuous.pdf The Continuous Infinitesimal Mathematics Philosophy JL Bell 著 - ?2005 Preface This book has a double purpose. First, to trace the historical development of the concepts of the continuous and the infinitesimal; and second, to describe the ways in which these two concepts are treated in contemporary mathematics. So the first part of the book is largely philosophical, while the second is almost exclusively mathematical. In writing the book I have found it necessary to thread my way through a wealth of sources, both philosophical and mathematical; and it is inevitable that a number of topics have not received the attention they deserve. Still, the thread itself, if tangled in places, has been luminous. “Only connect ... Live in fragments no longer,” says E. M. Forster, and that is what I have tried to do here. 0466現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 22:22:06.34ID:OrUOLzdR>>461-463
1.万に一つ、その定理と少なくとも証明が新しく、価値あるものなら、こんなところに書くのはもったいないよ(^^ 知り合いの数学科教官にでも見て貰って、投稿した方が良いぞ。 ここを見ている数学徒にしても、定理を引用しようとしたら、2CH(元5CH)では恰好悪いよ(^^ 2.見ていると思うが、無理数全体で微分可能な関数が出来ないことだけなら、解決済みだよ >>443に有るとおり ”THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, ・・・ on a co-meager set.” いままで読んだ範囲では、あなたのような定理は、使われいないようだ。 その定理が成立するなら、面白いと思うよ 3.ただ、面白い定理で価値あるなら、だれかがすでに書いている可能性もある (一方、少なくとも、自分はそれにはお目に掛かっていないので、新定理かも知れない) 0470132人目の素数さん2017/12/09(土) 23:19:30.90ID:B62Hdudt 阿呆スレ主があっという間に降参してワロタ
定理を引用しようとしたら、2CH(元5CH)では恰好悪いよ(^^ ↓ 定理を引用しようとしたら、2CH(現5CH)では恰好悪いよ(^^ 0472現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 23:24:03.24ID:OrUOLzdR>>470 "万に一つ、その定理と少なくとも証明が新しく、価値あるものなら、こんなところに書くのはもったいないよ(^^ ”(>>469) 0473現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 23:25:02.02ID:OrUOLzdR その定理が正しい確率を直観で表わしたんだが?(^^ 0474現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 23:32:59.79ID:OrUOLzdR>>443に有る(英文)定理では ”Let g be continuous and discontinuous on sets of points that are each dense in the reals. ”とある
1.Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.(>>285より) w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数) 無理数で0。ついでに、f_w(0) = 1 (>>285より。*) (「無理数で、リプシッツ連続」は>>284以下の既出文献でさんざん証明**)済みで略す) 2.f_w(p/q) = 1/w(q)>0と出来るとして、p/q(有理数)では、不連続になる。(自明だが念のために書いた) 3.このRuler Function に、新定理が適用可能とする。 4.R−B_f ⊂ Q = ∪[p∈Q] { p } …(1) (1)の右辺は疎な閉集合の可算和だから、上の新定理が使えて、f はある開区間(a,b)の上でリプシッツ連続になる。
? この後、そのままで良いのか?
特に、(a,b)の上で連続になる。QはR上で稠密だから、x∈(a,b)∩Qが取れる。 fは点xで不連続であるが、しかし(a,b)の上で連続に、矛盾する。 QED
a)なので、”このRuler Function に、新定理が適用可能”がおかしいか b)新定理がおかしいか
注 *)f_w(0) = 1を書く意味は、0は無理数でもなく、p/qとも表せないということかな **)(>>285より抜粋) ** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25) (引用終り)
http://mathforum.org/kb/message.jspa?messageID=5432910 Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007
これを読んでいて、疑問に思ったことが2点ある
1. ”[20] Marc Frantz, "Two functions whose powers make fractals", American Mathematical Monthly 105 #7 (Aug./Sept. 1998), 609-617. [MR 99g:28018; Zbl 952.28006] Following up on Darst/Taylor [18] above, Frantz investigates the Hausdorff dimension of the graphs of f^r.
THEOREM 1: If r > 2, then the Hausdorff dimension of the non-differentiability set for f^r is 2/r.”
一方、 ”[18] Richard Brian Darst and Gerald D. Taylor, "Differentiating Powers of an Old Friend", American Mathematical Monthly 103 #5 (May 1996), 415-416. [MR1400724; Zbl 861.26002]
Define f:R --> R by f(x) = 0 if x is irrational or zero, and f(p/q) = 1/q for p,q relatively prime with q > 0. They note that the set of points at which f is not continuous is the set of nonzero rational numbers.
THEOREM: If 1 < r <= 2, then f^r is differentiable only at zero. If r > 2, then f^r is differentiable almost everywhere (Lebesgue measure).”
だから、[18] からすると、If r > 2, then f^r is differentiable almost everywhere (Lebesgue measure).→Hausdorff dimension =1 で、"1 - 2/r(>>285)"ではないのでは?
2. ”Let g be continuous and discontinuous on sets of points that are each dense in the reals.” とは、continuous, discontinuous, 両者とも、Hausdorff dimension =1/2 見たいな形で、お互いが混じり合っているイメージなんだけど、おかしいかな? で、無理数と有理数だと、前者がHausdorff dimension =1、後者がHausdorff dimension =0 なんだけど・・・ 「函数の連続点の全体からなる集合は開集合の可算個の交わり(Gδ-集合)である。また不連続点の全体は閉集合の可算個の合併(Fσ-集合)である。」(by 上記wikipedia 不連続性の分類 ) だから、それで良いのか・・な(^^
Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.(>>285より) w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数)
は、おまえの新定理の反例になってないか?
1.(>>481 wikipediaより)「不連続点の全体は閉集合の可算個の合併(Fσ-集合)である」を認めるとする 2.”** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)”(>>285より) Hausdorff dimension zero → 個々の不連続点の閉集合は、R上長さを持たない、つまり、”内点を持たない”が言えると思う(未証明だが) 3.とすると、その定理の”R−B_f が高々可算無限個の疎な閉集合の和で被覆できる”が言えるだろ? 4.で、R−B_f は疎な閉集合の可算和だから、新定理が使えて、f はある開区間(a,b)の上でリプシッツ連続になる。 5.で、特に、(a,b)の上で連続になる。QはR上で稠密だから、x∈(a,b)∩Qが取れる。fは点xで不連続であるが、しかし(a,b)の上で連続に、矛盾する。
まあ、要するに、この”Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.”(>>285より)というのは ” be continuous and discontinuous on sets of points that are each dense in the reals.”(>>285より)が、実現された関数なわけだ
>1.(>>481 wikipediaより)「不連続点の全体は閉集合の可算個の合併(Fσ-集合)である」を認めるとする >2.”** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)”(>>285より) > Hausdorff dimension zero → 個々の不連続点の閉集合は、R上長さを持たない、つまり、”内点を持たない”が言えると思う(未証明だが) >3.とすると、その定理の”R−B_f が高々可算無限個の疎な閉集合の和で被覆できる”が言えるだろ?
1点、(>>497)”Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.(>>285より) w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数)” が、反例になるだろうと指摘した
1. (>>534 より)”1点、(>>497)”Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.(>>285より) w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数)” が、反例になるだろうと指摘した それ、>>284-285に出典が上がっているだろ? あなたのすべきことは、私への反論でなく、出典に当たって、本当に反例かそうでないか、直接確かめることじゃないのか?” と書いたけど、相変わらず原典に当たってないでしょ? あなたの反例でないという理屈が、単におれが挙げた理由付けについての反論に過ぎないでしょ?
それから、反例の原典に当たったと言うけれど、おれの言っている原典は、単に下記URLを覗くだけじゃなく、その元の引用文献に当たれってことだ! 例えば、下記[4][13]や、 H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function",Bulletin of the Calcutta Mathematical Society 49 (1957)
それに、[15] や、Edward Maurice Beesley, Anthony Perry Morse, and Donald Chesley Pfaff, "Lipschitzian points", American Mathematical Monthly 79 #6 (June/July 1972) くらいは、これは見とかないとね
(例えば、”(p. 373) "We omit the proof, because it is rather lengthy, and one would hope to generalize the theorem by replacing the rationals by an arbitrary dense set, and possibly to show that the set of points at which g fails to be Lipschitzian is a residual set."
NOTE: Sengupta/Lahiri had essentially obtained this result in 1957 (the points of discontinuity have to form an F_sigma set, however). See my remark in [13] above. This result is also proved in Gerald Arthur Heuer, "A property of functions discontinuous on a dense set", American Mathematical Monthly 73 #4 (April 1966), ”とかに関連した部分など。 あんたみたく、簡単に証明できるというなら、Heuer先生"We omit the proof, because it is rather lengthy”とは書かないだろう・・)
[13] Gerald Arthur Heuer, "Functions continuous at irrationals and discontinuous at rationals", abstract of talk given 2 November 1963 at the annual fall meeting of the Minnesota Section of the MAA, American Mathematical Monthly 71 #3 (March 1964), 349.
THEOREM: If g is continuous at the irrationals and not continuous at the rationals, then there exists a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
REMARK BY RENFRO: The last theorem follows from the following stronger and more general result. Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set). This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function", Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below.
[15] Gerald Arthur Heuer, "Functions continuous at the irrationals and discontinuous at the rationals", American Mathematical Monthly 72 #4 (April 1965), 370-373. [MR 31 #3550; Zbl 131.29201]
THEOREM 5: If g is a function discontinuous at the rationals and continuous at the irrationals, then there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
(p. 373) "We omit the proof, because it is rather lengthy, and one would hope to generalize the theorem by replacing the rationals by an arbitrary dense set, and possibly to show that the set of points at which g fails to be Lipschitzian is a residual set."
NOTE: Sengupta/Lahiri had essentially obtained this result in 1957 (the points of discontinuity have to form an F_sigma set, however). See my remark in [13] above. This result is also proved in Gerald Arthur Heuer, "A property of functions discontinuous on a dense set", American Mathematical Monthly 73 #4 (April 1966), 378-379 [MR 34 #2791]. Heuer proves that for each 0 < s <= 1 and for each f:R --> R such that {x: f is continuous at x} is dense in R and {x: f is not continuous at x} is dense in R, the set of points where f does not satisfy a pointwise Holder condition of order s is the complement of a first category set (i.e. a co-meager set). By choosing s < 1, we obtain a stronger version of Sengupta/Lahiri's result. By intersecting the co-meager sets for s = 1/2, 1/3, 1/4, ..., we get a co-meager set G such that, for each x in G, f does not satisfy a pointwise Holder condition at x for any positive Holder exponent. (Heuer does not explicitly state this last result.) A metric space version of Heuer's result for an arbitrary given pointwise modulus of continuity condition is essentially given in: Edward Maurice Beesley, Anthony Perry Morse, and Donald Chesley Pfaff, "Lipschitzian points", American Mathematical Monthly 79 #6 (June/July 1972), 603-608 [MR 46 #304; Zbl 239.26004]. See also the last theorem in Norton [17] below. (引用終り)
4.ところで、「一点におけるリプシッツ連続」については、”pointwise Lipschitz condition”という用語がある 例えば、>>285 "** For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition. Heuer [15]、** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]" とか 検索でも、pointwise Lipschitz condition で山ほどヒットするよ
[4] Bohus Jurek, "Sur la derivabilite des fonctions a variation bornee", Casopis Pro Pestovani Matematiky a Fysiky 65 (1935), 8-27. [Zbl 13.00704; JFM 61.1115.01]
It appears that Jurek proves some general results concerning the zero Hausdorff h-measure of sets of non-differentiability for bounded variation functions such that the sum of the h-values of the countably many jump discontinuities is finite (special case: h(t) = t^r for a fixed 0 < r < 1). General "h-versions" of the ruler function seem to appear as examples, and V. Jarnik's more precise results about the Hausdorff dimension of Liouville-like Diophantine approximation results are used. (引用終り)
そして、例の pdf の話に戻るが、この程度の pdf から逃げ回るなんて許さない。実質的には「補題1.5」と「定理1.7」しか 内容が無いシンプルな pdf なのだ。その部分は目測では2ページ分くらいしかない。>>621で指摘があった短縮案を加味すると、 さらにもう少し証明がシンプルになる。そのような、「たった2ページ」の証明から逃げ回るなんて言語道断である。 しかも、書いた本人がここに居て、何でも質問できるというのに。
に注意して、inf[δ>0] sup[0<|y−x|<δ]|(f(y)−f(x))/(y−x)|< N ということになるので、 あるδ>0に対して sup[0<|y−x|<δ]|(f(y)−f(x))/(y−x)|< N である。 ―――――――――――――――――――――――――――――――――――――――――――――――――
に注意して、inf[δ>0] sup[0<|y−x|<δ]|(f(y)−f(x))/(y−x)|< N ということになるので、 あるδ>0に対して sup[0<|y−x|<δ]|(f(y)−f(x))/(y−x)|< N である。 ――――――――――――――――――――――――――――――――――――――――――――――――― 補題1.5のうち、ここまでの議論については理解しているのか? YESかNOかで答えよ。NOの場合は、どこで躓いているのかも述べよ。 0650132人目の素数さん2017/12/13(水) 21:50:31.95ID:+Ojks0P8 يعقوب