現代数学の系譜工学物理雑談古典ガロア理論も読む47

[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47 (650ﾚｽ)
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441: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/09(土) 15:19:23.60 ID:OrUOLzdR

>>440 つづき

手短に要点を書くと、証明すべきことは・・・

＜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)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1512046472/441

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