現代数学の系譜工学物理雑談古典ガロア理論も読む56

[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56 (768ﾚｽ)
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287: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2018/12/23(日) 10:18:36.65 ID:aqLWE3+/

>>272
>”数学セミナー 2015年11月号 箱入り無数目 時枝 正の記事は誤り”は
>英語圏の数学常識として、知っておいた方が良いだろうから

＜補足追加しておく＞
１）追加１
http://forums.xkcd.com/viewtopic.php?t=103672
(抜粋)
A mind boggling riddle involving the Axiom of Choice
Postby Jrthedawg ≫ Thu Jul 18, 2013 7:47 pm UTC
Check out the claim made in this riddle/solution.
https://math.stackexchange.com/questions/371184/predicting-real-numbers (下記引用抜粋)
The claim is that equipped with the axiom of choice, these mathematicians can predict real numbers with 99% accuracy. That seems absurd. If I think of a real number between 0 and 1, measure theory says there is a 0% chance anyone can guess it. Why does this fail with the axiom of choice?
I'm confused.

Re: A mind boggling riddle involving the Axiom of Choice
by korona ≫ Sun Jul 21, 2013 6:43 pm UTC
The problem is that your intuitive notion of probability cannot be applied to sequences of real numbers. Probability is just not defined in this context.
As said above: Probability is not defined here. How did you get 10%?
(引用終り)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1544924705/287

288: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2018/12/23(日) 10:19:06.43 ID:aqLWE3+/

>>287

つづき
（上記引用はこれ）
https://math.stackexchange.com/questions/371184/predicting-real-numbers
(抜粋)
Predicting Real Numbers  May 15 '13 at 22:28 Jared
Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to see how others might think about it.
100 rooms each contain countably many boxes labeled with the natural numbers. Inside of each box is a real number. For any natural number n, all 100 boxes labeled n (one in each room) contain the same real number. In other words, the 100 rooms are identical with respect to the boxes and real numbers.
Knowing the rooms are identical, 100 mathematicians play a game.
99 out of 100 mathematicians must correctly guess their real number for them to (collectively) win the game.
What is a winning strategy?
(引用終り)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1544924705/288

289: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2018/12/23(日) 10:20:51.74 ID:aqLWE3+/

>>288

つづき

２）追加２
http://brainden.com/forum/topic/16510-100-mathematicians-100-rooms-and-a-sequence-of-real-numbers/
(抜粋)
100 mathematicians, 100 rooms, and a sequence of real numbers Asked by Jrthedawg, July 21, 2013
I am a collector of math and logic puzzles, and this must be the best I've ever seen.

100 rooms each contain countably many boxes labeled with the natural numbers. Inside of each box is a real number. For any natural number n, all 100 boxes labeled n (one in each room) contain the same real number. In other words, the 100 rooms are identical with respect to the boxes and real numbers.
Knowing the rooms are identical, 100 mathematicians play a game. After a time for discussing strategy, the mathematicians will simultaneously be sent to different rooms, not to communicate with one another again. While in the rooms, each mathematician may open up boxes (perhaps countably many) to see the real numbers contained within.
Then each mathematician must guess the real number that is contained in a particular unopened box of his choosing. Notice this requires that each leaves at least one box unopened.
99 out of 100 mathematicians must correctly guess their real number for them to (collectively) win the game.
What is a winning strategy?

つづく

http://rio2016.5ch.net/test/read.cgi/math/1544924705/289

290: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2018/12/23(日) 10:21:37.14 ID:aqLWE3+/

>>289
つづき

Rainman
Advanced Member
Posted July 31, 2013

Unfortunately foor the mathematicians, that doesn't work. But it is a good example illustrating the difference between arbitrarily large and infinite.
You are right that the mathematicians can choose a number N, and let each representative sequence begin with (1, 2, 3, ..., N). And you are right that there is no finite limit to how large N can be. They can make N arbitrarily large. But perhaps surprisingly, that doesn't mean they can go on like that infinitely.
They still must choose a number N, and there are no infinite numbers.
In terms of sequences, infinity means forever.
And forever means the sequence can't change at some point to match another sequence. There is only one sequence which goes x1=1, x2=2, x3=3, and so on for infinitely many terms.

As for the mathematicians, the unfortunate part is that no matter how large they make N, the probability is still 0 that their sequence matches its representative sequence at the N-th place. No matter how large they make N, it will still be infinitely small compared to the concept of infinity.
(引用終り)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1544924705/290

292: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2018/12/23(日) 10:23:37.78 ID:aqLWE3+/

>>291
つづき

４）まとめ
まあ、英語圏では、これは
・”Probability is not defined here”（確率は定義されない）
・”But it is a good example illustrating the difference between arbitrarily large and infinite. ”
　”As for the mathematicians, the unfortunate part is that no matter how large they make N, the probability is still 0 that their sequence matches its representative sequence at the N-th place.”
　（「arbitrarily large」が可能無限、「infinite」が実無限な ）
・”In this context, does it make sense to say "guess the content of a box with arbitrarily high probability"? I think it is ok, because the only probability measure we need is uniform probability on {0,1,…,N?1},
but other people argue it's not ok, because we would need to define a measure on sequences, and moreover axiom of choice messes everything up.”
　（英語圏でも、落ちこぼれは”I think it is ok”だけど・・、分っている人は”it's not ok, because we would need to define a measure on sequences, and moreover axiom of choice messes everything up.”）

つづく

http://rio2016.5ch.net/test/read.cgi/math/1544924705/292

320: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2018/12/23(日) 21:10:34.62 ID:aqLWE3+/

>>319

つづき

https://www.kaijo.ed.jp/
海城中学高等学校
https://www.kaijo.ed.jp/students?filter=true&genre=subject&genre_global&genre_subject=mathematics&post_date_from&post_date_to&post_tag&words&pg=4
数学科リレー講座 2013
４回目の今年は、「現代幾何学のひろがり」と題して、非ユークリッド幾何学をテーマの中心とします。
初日の今日は、今後6日間のガイダンスとして、毎日の聞き所をダイジェストで紹介。
https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-2_1.pdf
平成25年度 夏期講習 数学科リレー講座 2日目 第1部 非ユークリッド幾何
https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-3.pdf
2013 年 数学科リレー講座. 3 日目：球面幾何学

余録
https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-1.pdf
平成 25 年度 数学科夏期リレー講座「現代幾何学のひろがり」

https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-4.pdf
2013年度 数学科リレー講座 4日目 ?非ユークリッド幾何学の例

https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-5_2.pdf
ミンコフスキー幾何 5日目 - 海城中学高等学校

https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-6.pdf
平成 25 年度数学科リレー講座. 6 日目. エルランゲン・プログラム

https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2013summer-7.pdf
2013年度 数学科リレー講座 状況

https://www.kaijo.ed.jp/wp-content/uploads/2016/02/2012summer_5Kawasaki.pdf
2012年度・夏期数学科リレー講座・5日目 “リーマン面 ... - 海城中学高等学校

以上

http://rio2016.5ch.net/test/read.cgi/math/1544924705/320

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