現代数学の系譜工学物理雑談古典ガロア理論も読む49

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む49 (658ﾚｽ)
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40: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/28(木) 23:41:49.82 ID:IsA0R4yK

>>30
沢山のレスがありがとう
まあ、ゆっくりやろう

まだ、疑問に思っているのは
下記のDifferentiability of the Ruler Functionの記述と貴方の定理との整合性だ

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
(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.

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.)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1514376850/40

366: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/10(水) 21:42:07.86 ID:xixJS48Q

>>364 追加

ちょっと思いついたので、悪いが、忘れないうちに下記を書いておく

（>>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）

簡単のために、区間[0, 1]を考える。（同じことを、区間[n, n+1] (ｎは整数)で考えれば、実数R全体に展開できる）

このような、場合、上記数学者のRenfroさんや、Robertsさんたちは、”Qで不連続、リュービル数（超越数）で微分不可（リプシッツ連続でもない）だが、それ以外の無理数では、微分可だ”という

つづく

http://rio2016.5ch.net/test/read.cgi/math/1514376850/366

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