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)
