High level people は自分達で勝手に立てたスレ28へどうぞ！sage進行推奨（＾＾； また、スレ43は、私が立てたスレではないので、私は行きません。そこでは、私はスレ主では無くなりますからね。このスレに不満な人は、そちらへ。 http://rio2016.2ch.net/test/read.cgi/math/1506152332/ 旧スレが512KBオーバー（又は間近）で、新スレ立てる （スレ主の趣味で上記以外にも脱線しています。ネタにスレ主も理解できていないページのURLも貼ります。関連のアーカイブの役も期待して。）
（>>40より） http://mathforum.org/kb/message.jspa?messageID=5432910 （>>35より） Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 より The modefied ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and f(x) = 1/w(q) if x = p/q ∈Q where p and q are relatively prime integers with q > 0.
ここに w(q)：an increasing function that eventually majorizes every power function. （w(q)は、どんなｐの冪より早く増大する関数 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. などではP532で、” (e.g., ai = 1/i^(i^i) )”などと記されている。qで書けば、= 1/q^(q^q)だ）
The modefied ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and (さらに有理数で場合けして) f(x) = 0 if q> m, x = p/q ∈Q f(x) = 1/w(q) if q<=m, x = p/q ∈Q where p and q are relatively prime integers with q > 0. また、他の条件は、すべて上記に同じ
さて、f(x) = 0 if x is irrational→f(x) = F(x) if x is irrationalとする
The modefied ruler function f is defined by f(x) = F(x) if x is irrational, f(0) = 1, and (さらに有理数で場合けして) f(x) = F(x) if q>= m, x = p/q ∈Q f(x) = F(x)+ 1/w(q) if q< m, x = p/q ∈Q where p and q are relatively prime integers with q > 0.
The modefied ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and (さらに有理数で場合けして) f(x) = 0 if q>= m, x = p/q ∈Q f(x) = 1/q^ν if q< m, x = p/q ∈Q where p and q are relatively prime integers with q > 0. ここに、ν>=0の実数とする
この場合も、mが有限の値の場合、不連続点は、分母q がある値ｍ以下の場合のみの有限個になる この場合も、「定理1.7 (422 に書いた定理)」が常に成り立ち ”R－Bf が内点を持たない閉集合の（有限個の）可算和で被覆でき、 f はある開区間の上でリプシッツ連続である．”となる
（>>376に書いた通りだが） （>>180）”定理1.7 (422 に書いた定理) f : R → R とする． Bf :={x ∈ R ｜ lim sup y→x ｜（f(y) － f(x)）/（y － x）｜< +∞ } と置く: もしR－Bf が内点を持たない閉集合の高々可算和で被覆できるならば、 f はある開区間の 上でリプシッツ連続である．”
で、 １）補集合R－Bfが、”が内点を持たない閉集合の可算”有限”和で被覆できるならば、 f はある開区間の上でリプシッツ連続である．”は、正しい しかし ２）補集合R－Bfが、”が内点を持たない閉集合の”稠密”分散可算無限和で被覆できるならば、 f はある開区間の上でリプシッツ連続とはできない．”が、正しい
中学校レベルの話だろう 0383１３２人目の素数さん2018/01/11(木) 10:14:50.34ID:pCvZqe21>>375 >さて、f(x) = 0 if x is irrational→f(x) = F(x) if x is irrationalとする > >The modefied ruler function f is defined by >f(x) = F(x) if x is irrational, >f(0) = 1, and >(さらに有理数で場合けして) >f(x) = F(x) if q>= m, x = p/q ∈Q >f(x) = F(x)+ 1/w(q) if q< m, x = p/q ∈Q >where p and q are relatively prime integers with q > 0. > >ここに、 F(x) は、簡単のために、解析函数で多くの多項式や初等関数のように、 >無限大のみに極を持つとする。（有限の範囲に極があっても問題ないが、記述が複雑になる） >>367の補足としてmを或る値として続けて書いているようだが、 解析関数 F(x) の定義域は複素平面Cの弧状連結で開円盤を含む開集合だから、 虚部が0ではない何らかの複素数xに対しても F(x) は定義されることになる。 だが、f(x)＝F(x) としているのに、そのような複素数に対する F(x) の複素数値の定義がどこにもなされていないので、 その定義は意味をなさない。結局実関数 f(x) を直線R上で定義することになる。 0384１３２人目の素数さん2018/01/11(木) 10:33:48.09ID:pCvZqe21>>381 トマエ関数や modefied ruler function は有理数か無理数かが分かってからその関数値が決まるような関数。 で、有理数全体Qは直線Rの部分空間としてのルベーグ測度が0の可測空間で、無理数の全体 R＼Q はその補集合になる上、 有理数に対して決まるそれらの関数値の決まり方は定義域Rの点としての既約分数 p/q の分母qや分子pの値にもよるから、 そういった関数を持ち出して有理数か無理数かを基準にして考えても何の意味もない。 0385現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/11(木) 10:54:01.96ID:clSPRjXH>>383 >結局実関数 f(x) を直線R上で定義することになる。
ところで、下記にen.wikipediaにいろいろあるが コーシーは、７番目”7.Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules.” やね
https://en.wikipedia.org/wiki/Rigidity_(mathematics) Rigidity (mathematics) (抜粋) In mathematics, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect.
The above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians.
Some examples include:
1.Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. 2.Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem.
3.By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set, say N, or the unit disk. By the previous example, a polynomial is also determined within the set of holomorphic functions by the finite set of its non-zero derivatives at any single point. 4.Linear maps L(X, Y) between vector spaces X, Y are rigid in the sense that any L ∈ L(X, Y) is completely determined by its values on any set of basis vectors of X. 5.Mostow's rigidity theorem, which states that the geometric structure of negatively curved manifolds is determined by their topological structure. 6.A well-ordered set is rigid in the sense that the only (order-preserving) automorphism on it is the identity function. Consequently, an isomorphism between two given well-ordered sets will be unique. 7.Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules. 8.Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface.
See also Uniqueness theorem Structural rigidity, a mathematical theory describing the degrees of freedom of ensembles of rigid physical objects connected together by flexible hinges. This article incorporates material from rigid on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. (引用終り)
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis. (引用終り)
https://en.wikipedia.org/wiki/Rigidity Rigidity (抜粋) Mathematics and physics ・Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity ・Structural rigidity, a mathematical theory of the stiffness of ensembles of rigid objects connected by hinges ・Rigidity (electromagnetism), the resistance of a charged particle to deflection by a magnetic field ・Rigidity (mathematics), a property of a collection of mathematical objects (for instance sets or functions) ・Rigid body, in physics, a simplification of the concept of an object to allow for modelling ・Rigid transformation, in mathematics, a rigid transformation preserves distances between every pair of points
Other uses ・Real rigidity, and nominal rigidity, the resistance of prices and wages to marketchanges in macroeconomics ・Ridgid, a brand of tools (引用終り)
”1Source unknown. I heard it from Benjy Weiss, who heard it from ..., who heard it from ... . For a related problem, see http://xorshammer.com/2008/08/23/set-theory-and-weather-prediction/ ” （上記URLは、関数数当てパズルで ”For some interesting comments on this puzzle, see Greg Muller’s blog post on it here and Chris Hardin and Alan Taylor’s paper An Introduction to Infinite Hat Problems.”とある）
関数数当てパズルの元のAlan D. Taylor さんの２つの論文とそのPDFリンク 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.
pdf：A peculiar connection between the Axiom of Choice and predicting the future THE MATHEMATICAL ASSOCIATION OF AMERICA Monthly February 2008 については、当時哲学者がいろいろ議論したらしい（下記のpdfご参照） だが、数学者の投稿は見つからなかった！！（＾＾
そして、過去スレ47にも書いたが、The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems (Developments in Mathematics) 2013 edition by Hardin, Christopher S., Taylor, Alan D. (2013) （>>439） では、上記の未来予測可能とか、任意の関数の値が予測可能とする論は、全部捨てられている
その話も、ちょろっと、まとめPDFに入れて貰えると面白いと思うよ で、時枝も同じだよ
https://link.springer.com/chapter/10.1007/978-3-319-58507-9_10 Philosophical Aspects of an Alleged Connection Between the Axiom of Choice and Predicting the Future, Pawel Pawlowski First Online: 06 September 2017 Abstract In 2008 Christopher Hardin and Alan Taylor published an article titled “Peculiar connection between the axiom of choice and predicting the future” in which they claim that if some system can be described as a function from a set of some instants of time to some set of states, then there is a way to predict the next value of the function based on its previous input. Using their so-called μμ -strategy one can randomly choose an instant t and the probability that the strategy is correct at t (i.e. that the output for a strategy for input t is exactly the same as the value of the function) equals 1. Mathematical aspects of this article are sound, but the background story about the correlation between theorems and philosophical aspects of predicting the future faces certain problems. The goal of my paper is to bring them up.
[PDF]A Proof of Induction? https://quod.lib.umich.edu/cgi/p/pod/dod-idx/proof-of-induction.pdf?c=phimp;idno=3521354.0007.002 philosophers’imprint A George 著 Department of Philosophy Amherst College- ?2007 - ?被引用数: 10 - ?関連記事 “Hardin-Taylor rule”, I shall call it ? that will, for any arbitrarily chosen function f, correctly predict most values of f on the basis of its past be- havior; that is, for most t the rule will correctly predict f (t) on the basis of f 's values at all s < t. I shall first sketch the proof's central idea and then turn to assess the result's philosophical significance. We begin by well-ordering R, the set of all functions from R to. R. (Here the Axiom of Choice must be employed, a fact to which we shall return.) That is ...
[PDF]Justifying Induction Mathematically: Strategies and Functions http://media.philosophy.ox.ac.uk/assets/pdf_file/0003/36651/LogiqueetAnalyseFinal08.pdf A PASEAU 著 Logique & Analyse 203 (2008), 263?269 - ?被引用数: 2 - ?関連記事 2008/08/27 - page 265 i i i i i i i i. JUSTIFYING INDUCTION MATHEMATICALLY: STRATEGIES AND FUNCTIONS. 265. These objections are answerable to a degree. The use of the Axiom of. Choice is indeed essential; but these days the ... Hardin-Taylor proof as providing a reliable present-predicting strategy. Once it is appreciated, the Hardin-Taylor proof can no longer plausibly be called a predictive strategy. Because it is so nonconstructive, it fails to yield a strat- egy.
>一つの根拠が、Chris Hardin and Alan Taylor’s paperに行き着く >だが、これが間違いだったと、彼らが自分達が後の論文で訂正しているよ（参考>>446-447） >それは、過去すれ47に書いた
これだな 現代数学の系譜 工学物理雑談 古典ガロア理論も読む47 https://rio2016.5ch.net/test/read.cgi/math/1512046472/171 (抜粋) 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（＾＾ (引用終り) 0454現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/13(土) 12:01:52.77ID:rUYSYDib>>452 問題読んで無いだろ？（＾＾ 0455１３２人目の素数さん2018/01/13(土) 12:14:07.95ID:ZJErTONp>>454 スレ主が以前ここに時枝記事をコピペした。(このコピペのことを忘れたとはいうべきではない) それを読むと、時枝記事には曖昧に書かれている部分があって、その部分を理解して把握することに時間がかかった。 時枝の確率の問題についても同じ状況だった。このスレではそのような状況だった。 やがて、正確に時枝記事を理解し把握した後は、標本空間が有限集合であるような確率の問題であることが分かった。 標本空間が有限集合であるような確率の問題は、高校どころか中学の確率の問題だ。 0456１３２人目の素数さん2018/01/13(土) 13:16:09.57ID:p9CVPkNb>>454 一年生用教科書読んで無いだろ？ 0457１３２人目の素数さん2018/01/13(土) 14:01:52.64ID:baaiEdIz 実数の連続性が分からない人って論理だけでは創られ ていない数学を論理だけで理解しているのだろうか。 距離空間の完備性が分からない人にＲの連続性と本質 的に同じことを説明したら理解されたんだけど中間値 の定理や最大値の定理を何も見ないで証明できるくら いの人が実数の連続性が分からないって 0458◆QZaw55cn4c 2018/01/13(土) 14:20:18.80ID:zUI9hxZm>>457 いや今それで悩んでいるのです 実数の完備のみならず一般の代数系での完備となると、なかなか理解がおよびません 可換な半群 L が演算子ρのもとで完備な順序集合のとき二元 a b ∈L の上限を aρb とすれば L はρについて半束になる…うーん 0459１３２人目の素数さん2018/01/13(土) 15:57:24.33ID:BB1mEg7b 藁 0460１３２人目の素数さん2018/01/13(土) 19:11:43.62ID:sUwT3lGp 有理数に入り切らない数 0461現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/13(土) 21:15:46.03ID:rUYSYDib>>455 おっちゃんな、スレ主だけど 原文をきちんと読むべきと思うよ
