現代数学の系譜工学物理雑談古典ガロア理論も読む48

[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48 (625ﾚｽ)
上下前次1-新
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

42: 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/15(金) 07:33:53.86 ID:dUFtnfpO(3/14) AAS
>>39
高校生かい

その話なら、こちらの方（下記）が良いよ
もっと良いのは、学校の先生に聞くことだな
５CH数学板は、高校生の勉強には向かないよ（＾＾

分からない問題はここに書いてね438
2chｽﾚ:math

48: １３２人目の素数さん [] 2017/12/15(金) 08:33:50.86 ID:esELFcoa(2/3) AAS
スレ主はεδを理解していないだけでなくそれが致命的であることも理解していない
教科書を一度もまともに勉強したことはバレバレだ

83(2): １３２人目の素数さん [] 2017/12/16(土) 10:55:38.86 ID:Ysy8e8oN(1) AAS
［定理］
開集合Ωに含まれる任意のコンパクト集合で値が一致
するふたつ関数はΩ全体で一致する
［証明］
Ωに含まれる任意のコンパクト集合Kに対してΩ＼Kで
値が一致しないとしてもKの取り方は任意だからKよ
り大きいΩに含まれるコンパクト集合K'が存在してK'
で値が一致することになるが(Ω＼K)∩K'では値が一致
せずかつ値が一致するからΩ＼Kで値が一致しないこ
とはなくΩ全体で値が一致する［証明終了］

［解説］
要は任意に与えられたΩに含まれるコンパクト集合K
に属さないΩの点xで値が一致しないとしてもKの取り
方は任意だからxを含むようにKを取り直せばKにおい
て値が一致するからΩの点で値が一致しない点は存在
しないということ

111(3): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/16(土) 14:37:26.86 ID:/2xvBEHK(25/58) AAS
>>110 つづき

[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.

The complete text of the abstract follows, with minor editing changes to accommodate ASCII format.

Earlier results of Porter, Fort, and others suggest additional questions about the functions in the title. Differentiability and Lipschitz conditions are considered. Special attention ispaid to the ruler function (f) and its powers.
Sample results:
THEOREM:
If 0 < r < 2, f^r is nowhere Lipschitzian; f^2 is nowhere differentiable, but is Lipschitzian on a dense subset of the reals.
THEOREM:
If r > 0, f^r is continuous but not Lipschitzian at every Liouville number;
if r > 2, f^r is differentiable at every algebraic irrational.
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.

つづく

118(1): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/16(土) 15:04:54.86 ID:/2xvBEHK(31/58) AAS
>>116
>"c points"の意味が分らん（＾＾

違うかも知れないが、検索ヒットと他にめぼしいヒットがないので下記を貼る
（下記だと、cは連続濃度の意味だね）
https://mathoverflow.net/questions/102386/is-a-random-subset-of-the-real-numbers-non-measurable-is-the-set-of-measurable
(抜粋)
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
edited Nov 29 '12 at 22:06

19 answered Jul 16 '12
The answer to your second question (assuming the axiom of choice, to dodge Asaf's comment) is that 2^R/Σ has dimension 2^c, where c=2^?0 is the cardinality of the continuum.
The main ingredient of the proof is a partition of [0,1] into c subsets, each of which intersects every uncountable closed subset of [0,1].
To get such a partition,
first note that there are only c closed subsets of [0,1], so you can list them in a sequence of length (the initial ordinal of cardinality) c in such a way that each closed set is listed c times.
Second, recall that every uncountable closed subset of [0,1] has cardinality c.
Finally, do a transfinite inductive construction of c sets in c steps as follows:
At any step, if the closed set at that position in your list is C and if this is its α-th occurrence in the list,
then put an element of C into the α-th of the sets under construction, being careful to use an element of C that hasn't already been put into another of the sets under construction.
You can be this careful, because fewer than c points have been put into any of your sets in the fewer than c preceding stages, while C has c points to choose from. At the end, if some points in [0,1] remain unassigned to any of the sets under construction, put them into some of these sets arbitrarily, to get a partition of [0,1].

つづく

128: 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/16(土) 16:22:20.86 ID:/2xvBEHK(37/58) AAS
>>127 つづき

（参考）
スレ46 2chｽﾚ:math
＜引用＞
http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION,
DIOPHANTINE APPROXIMATION,
AND A REFORMULATION
OF THE THUE-SIEGEL-ROTH THEOREM
JUAN LUIS VARONA
This paper has been published in Gazette of the Australian Mathematical Society, Vol-
ume 36, Number 5, November 2009, pp. 353{361.
Received 29 February 2008; accepted for publication 6 October 2009.
(抜粋)

ここに
fν(x)
＝0　if x ∈ R - Q（無理数）
＝1/q^ν if x = p/q ∈ Q, irreducible （有理数で既約分数）
で

Theorem 1. For ν > 2, the function fν is discontinuous (and consequently not differentiable) at the rationals, and continuous at the irrationals.
With respect the differentiability, we have:
(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.
(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.
Moreover, the sets of numbers that fulfill (a) and (b) are both of them un-countable.
(引用終り)

以上

178(1): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/17(日) 10:11:58.86 ID:uVIGteN6(12/26) AAS
まあ、そりゃー、プロ数学界の多数が「成立」と保証している定理の証明は別だ。だから、「読め読め」というなら、プロ数学者のお墨付きをどうぞ

186(3): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/17(日) 13:19:36.86 ID:uVIGteN6(14/26) AAS
>>179-180
>とでも言うのであろう。そもそも、「2ページの証明」という超手軽な分量の時点で、
>その証明「を読まない」という理屈は全く通用しない。

お断りだな
数学史を学べば、過去多くの天才と呼ばれる大数学者が正しいと、周囲も一度は認めた証明に、穴があったという事例は多い

つづく

240(1): １３２人目の素数さん [] 2017/12/18(月) 20:29:11.86 ID:OYSP7g0j(1/2) AAS
スレ主　国語　国語

508(1): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/24(日) 21:03:53.86 ID:Q5UHveEY(13/18) AAS
>>505
ID:ndfap2+Cさん、どうも。スレ主です。

レスありがとう（＾＾
あなたは
(>>401)
"どっちもどっち
ID:KNjgsEZnはただの基地外"
と書いてくれた人かな？

>>・いま、考えている通常のRにおいて、「内点を持たない閉集合」とは、孤立する１点から成る集合にほかならない
>違うよ
>境界点だけということ
>境界点が孤立してなくてはいけないわけではない

ああ、あなたは、レベル高そうだな（＾＾
ちょっと考えてみるよ

533: 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/25(月) 17:46:20.86 ID:I8rwcj5/(2/3) AAS
>>531
「ぷふ」さん、お元気そうでなによりです！＼（＾＾／

「ぷふ」さんには、感謝していますｍ（＿＿）ｍ

例の時枝に、終止符を打っていただいて（＾＾

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.047s