純粋・応用数学・数学隣接分野（含むガロア理論）15

[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）15 (1002ﾚｽ)
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560: １３２人目の素数さん [] 2023/08/15(火) 20:08:33.72 ID:TPe4RI1p

>>599
では、代用で 検索キーワード
google book The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems Christopher S. Hardin , Alan D. Taylor
これで、下記google bookがヒット（後に Gabay で検索 13件）
私の結論：下記はあくまでHat Problemsの本で、「箱入り無数目」を正当化できるものにあらず！

https://books.google.co.jp/books/about/The_Mathematics_of_Coordinated_Inference.html?id=69E-AgAAQBAJ&redir_esc=y
The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems
Christopher S. Hardin, Alan D. Taylor
Springer Science & Business Media, 2013/10/17 - 109 ページ

Two prisoners are told that they will be brought to a room and seated so that each can see the other. Hats will be placed on their heads; each hat is either red or green. The two prisoners must simultaneously submit a guess of their own hat color, and they both go free if at least one of them guesses correctly. While no communication is allowed once the hats have been placed, they will, however, be allowed to have a strategy session before being brought to the room. Is there a strategy ensuring their release? The answer turns out to be yes, and this is the simplest non-trivial example of a “hat problem.”

This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domain based on some knowledge of its values at other points. Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics. For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a – 1, a), and for every such function the prediction is incorrect only on a countable set that is nowhere dense.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1691376403/560

561: １３２人目の素数さん [] 2023/08/15(火) 20:08:59.23 ID:TPe4RI1p

つづき

The monograph progresses from topics requiring fewer prerequisites to those requiring more, with most of the text being accessible to any graduate student in mathematics. The broad range of readership includes researchers, postdocs, and graduate students in the fields of set theory, mathematical logic, and combinatorics. The hope is that this book will bring together mathematicians from different areas to think about set theory via a very broad array of coordinated inference problems.

目次
Chapter 1 Introduction 1
Chapter 2 The Finite Setting 10
Full Visibility 19
OneWay Visibility 29
Chapter 5 Dual Hat Problems Ideals and the Uncountable 48
Neutral and Anonymous Predictors 61
Chapter 7 The Topological Setting 70
Chapter 8 Universality of the μPredictor 83
Chapter 9 Generalizations and Galois Tukey Connections 93
Bibliography 103

キーワード Gabay で検索 13件
Springer Science & Business Mediaの許可を受けてページを表示しています.

1)7 ページ
But something quite unexpected also turns out to be true, as was noticed by Yuval Gabay and Michael O'Connor while they were graduate students at Cornell in 2004.

2)19 ページ
There is a predictor ensuring that all but finitely many—not just infinitely many—are correct, and this is what Gabay and O'Connor obtained using the axiom of choice. Such a predictor is called a finite-error predictor.

3)20 ページ
In Sect.3.2, we present the Gabay-O'Connor theorem and we point out the essential uniqueness of their predictor. In Sect. 3.3, we derive from this a result of Hendrik Lenstra that guarantees a predictor ensuring that every agent will ...

つづく

http://rio2016.5ch.net/test/read.cgi/math/1691376403/561

562: １３２人目の素数さん [] 2023/08/15(火) 20:10:04.56 ID:TPe4RI1p

つづき

4)21 ページ
The proof of the Gabay-O'Connor theorem above naturally extends to the following more general form. (Recall the definition of product filter from page 4.) Theorem 3.2.3 (Gabay, O'Connor). Let I be an ideal on A, and suppose we have a ...

5)22 ページ
Intuitively, the strategy is for everyone to take it on faith that the (finite) group sum of the differences between the true coloring and the guesses provided by the Gabay-O'Connor theorem is the identity of the group.

6)24 ページ
... S0.h// D h.a/, as desired. ut For Gabay's extension of the hat problem in Sect.2.4, suppose there are countably many agents labeled by natural numbers and lined up in order so that everyone sees the hats of higher-numbered agents, ...

7)25 ページ
We derived Lenstra's theorem from the Gabay-O'Connor theorem, so Theorem 3.4.2 also shows us that some nontrivial version of the axiom of choice is needed to prove the Gabay-O'Connor theorem. However, the Gabay-O'Connor theorem, ...

8)26 ページ
Theorems 3.4.2 and 3.4.3 can be recast in the context of Lebesgue measurability, to show that Lenstra's theorem and the Gabay-O'Connor theorem both imply the existence of non-measurable sets of reals. However, to show that ZF C DC ...

9)27 ページ
Is the Gabay-O'Connor theorem equivalent to the axiom of choice? We also don't know the answer to the following question about the partition relations that we used in this chapter. Question 3.6.2. Does a -> [o]' imply ad -(a))''' ...

つづく

http://rio2016.5ch.net/test/read.cgi/math/1691376403/562

563: １３２人目の素数さん [] 2023/08/15(火) 20:10:24.40 ID:TPe4RI1p

つづき

10)59 ページ
A Study of Generalized Hat Problems Christopher S. Hardin, Alan D. Taylor. the whole coloring correctly. This observation essentially originated with Gabay and O'Connor. 5.6. Open. Questions. First, as pointed out in Sect.

11)62 ページ
For any a>m, we have f #a D x#.n C a m/, and agent a will guess p.f#a/ D xnCam D fmCam D fa, so a guesses correctly. ut If the set of agents is the set of natural numbers, then Galvin's theorem is stronger than the Gabay-O'Connor ...

12)75 ページ
So, since f and hfiz0 agree on U0fz0g, guesses correctly at z1, a contradiction. ut. 7.3. Corollaries. The easy generalization of the Gabay-O'Connor theorem to the context of ideals that is described at the ...

13)106 ページ(Indexの部分らしい)
M/, 35 D E e, 45 Ebert, Todd, 1 EO, see parity relation Erd ̋os, Paul, 64 error set, 72 evading family, 45 evasion number, 45 F Feige, Uriel, 14 filter, 4 non-principal, 4 product, 4 proper, 4 G Gabay, Yuval, 7, 19, 22 Gabay-O'Connor ...
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1691376403/563

572: １３２人目の素数さん [] 2023/08/15(火) 23:48:13.96 ID:TPe4RI1p

>>568
>there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a – 1, a)
>とありますけど？

なるほど
ありがとう
これ、謎のプロ数学者氏が好きそうな話題かもねｗ

えーと>>560より、関連含め引用下記
This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more points of its domain based on some knowledge of its values at other points.
Topics range from hat problems that are accessible to everyone willing to think hard, to some advanced topics in set theory and infinitary combinatorics.

For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a – 1, a), and for every such function the prediction is incorrect only on a countable set that is nowhere dense.
ですね
＜最後の行のgoogle訳(一部手直し)＞
たとえば、開区間 (a – 1, a) 上の f の値の知識のみに基づいて、実数を実数にマッピングする関数 f の値 f(a) を予測する方法があり、そのような関数ごとに 予測が正しくないのは、密度がどこも稠密でない可算集合の場合のみです。

これで私の解釈は
１）普通は、開区間 (a – 1, a)は、いわゆる開集合で、この区間の実数の連続濃度の関数値を想定している
２）開区間 (a – 1, a)上の連続濃度の関数値 f(a-1)〜f(a)が知られている
３）それを利用して、f(a)の値を推定する。これは、普通”外挿”と呼ばれる手法です
４）もし、fが連続関数ならば、極限を使って、下からaに近づくようにして
　　lim x→a {f(x)} とすれば、 f(x)→f(a)ですよね？
５）「予測が正しくないのは、密度がどこも稠密でない可算集合の場合のみ」
　この意味は、開区間 (a – 1, a)の関数値の情報が、"稠密でない可算集合"つまり、離散値と解せられます
６）ともかくも、「箱入り無数目」は、適用外ですね
７）fが不連続関数なら？ さあ？ 本を読まないと分かりませんね

http://rio2016.5ch.net/test/read.cgi/math/1691376403/572

583: 新 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/08/16(水) 13:02:48.72 ID:VDPkTO11

>>572 追加
　>>560より
https://books.google.co.jp/books/about/The_Mathematics_of_Coordinated_Inference.html?id=69E-AgAAQBAJ&redir_esc=y
ここで、内部を検索で、キーワード predict
で、下記2件 ヒット

vii ページ
This book deals with the question of how successfully one can predict the value of an arbitrary function at one or more ... For example, there is a method of predicting the value f.a/ of a function f mapping the reals to the reals, ...

8 ページ
In Chap.7 we start to move further away from the hat problem metaphor and think instead of trying to predict a function's value at a point based on knowing (something about) its values on nearby points ...

そして 560-561より
Chapter 7 The Topological Setting 70
だったから、ここらから
”For example, there is a method of predicting the value f(a) of a function f mapping the reals to the reals, based only on knowledge of f's values on the open interval (a-1, a), and for every such function the prediction is incorrect only on a countable set that is nowhere dense.”
についての説明があるのですね

そして、明らかに、開区間 (a-1, a)の連続濃度の関数値を使って、開区間外の点aの関数値 f(a)を推定する話
外挿ですね

つづく

http://rio2016.5ch.net/test/read.cgi/math/1691376403/583

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