https://ja.wikipedia.org/wiki/%E5%BD%A2%E5%BC%8F%E7%9A%84%E5%86%AA%E7%B4%9A%E6%95%B0 形式的冪級数 (抜粋) 定義 A を可換とは限らない環とする。A に係数をもち X を変数(不定元)とする(一変数)形式的冪級数 (formal power series) とは、各 ai (i = 0, 1, 2, …) を A の元として、 Σn=0〜∞ anX^n=a0+a1X+a2X^2+・・・ の形をしたものである。ある m が存在して n >= m のとき an = 0 となるようなものは多項式と見なすことができる。 形式的冪級数全体からなる集合 A[[X]] に和と積を定義して環の構造を与えることができ、これを形式的冪級数環という。
性質 ・多項式とは異なり、一般には、「代入」は意味を持たない。無限個の和が出てきてしまうからである。 しかし、例えば次のようなときには意味を持つ。可換環 A はイデアル I による I 進距離で完備であるとする。 このとき a1,・・・ ,an∈ I であれば、 Σ α cα X^α ∈ A[[X1,・・・ ,Xn]] の X1,・・・ ,Xn に a1,・・・ ,an を代入したものは収束する。 (引用終り) 以上 0165現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/07(火) 21:16:43.00ID:b2sufDWR>>161 おい、ムリしなくて良いぞ お前の頭じゃ、理解できないんだろ?
”the emphasis on the types ("species") of objects”(下記)って、なんですかね? あんまし、説得力ないと思う それより、圏論的説明をしっかりやるべきでは?(^^; ("species"って、マイナーな印象しか受けない。数学のメインストリームじゃないでしょ?)
https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory Inter-universal Teichmuller theory (抜粋) History In March 2018, Peter Scholze and Jakob Stix visited Kyoto University for five days of discussions with Mochizuki and Yuichiro Hoshi; while this did not resolve the differences, it brought into focus where the difficulties lay.[8][10]
In September 2018, Mochizuki wrote a 41-page summary of his view of the discussions and his conclusions about which aspects of his theory he considers misunderstood.[12] In particular he names: ・"re-initialization" of (mathematical) objects, making their previous "history" inaccessible; ・"labels" for different "versions" of objects; ・the emphasis on the types ("species") of objects.
http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf IUTそのIV (抜粋) P68 In the following discussion, we use the phrase “set-theoretic formula” as it is conventionally used in discussions of axiomatic set theory [cf., e.g., [Drk], Chapter 1, §2], with the following proviso: In the following discussion, it should be understood that every set-theoretic formula that appears is “absolute” in the sense that its validity for a collection of sets contained in some universe V relative to the model of set theory determined by V is equivalent, for any universe W such that V ∈ W, to its validity for the same collection of sets relative to the model of set theory determined by W [cf., e.g., [Drk], Chapter 3, Definition 4.2]. Definition 3.1. (i) A 0-species S0 is a collection of conditions given by a set-theoretic formula P85 Bibliography [Drk] F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974).
species って、nLabでは圏論なんだけど 望月 IUT4 §3では、ZFCの集合論みたく書いてある はて はて? (^^;
https://ncatlab.org/nlab/show/species nLab species (抜粋) 1. Idea A (combinatorial) species is a presheaf or higher categorical presheaf on the groupoid core(FinSet), the permutation groupoid.
A species is a symmetric sequence by another name. Meaning: they are categorically equivalent notions.
2. Definition 1-categorical
2-categorical
(∞,1) -categorical
Operations on species There are in fact 5 important monoidal structures on the category of species. 0176現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/08(水) 12:15:31.15ID:1QCooAdl>>175 追加
IUTその4に下記説明ある が、圏論で筋通した方が良さそう?
P72 Example 3.2. Categories. The notions of a [small] category and an isomorphism class of [covariant] functors between two given [small] categories yield an example of a species. That is to say, at a set-theoretic level, one may think of a [small] category as, for instance, a set of arrows, together with a set of composition relations, that satisfies certain properties; one may think of a [covariant] functor between [small] categories as the set given by the graph of the map on arrows determined by the functor [which satisfies certain properties]; one may think of an isomorphism class of functors as a collection of such graphs, i.e., the graphs determined by the functors in the isomorphism class, which satisfies certain properties. Then one has “dictionaries” 0-species ←→ the notion of a category 1-species ←→ the notion of an isomorphism class of functors at the level of notions and a 0-specimen ←→ a particular [small] category a 1-specimen ←→ a particular isomorphism class of functors at the level of specific mathematical objects in a specific ZFC-model. Moreover, one verifies easily that species-isomorphisms between 0-species correspond to isomorphism classes of equivalences of categories in the usual sense.
Remark 3.2.1. Note that in the case of Example 3.2, one could also define a notion of “2-species”, “2-specimens”, etc., via the notion of an “isomorphism of functors”, and then take the 1-species under consideration to be the notion of a functor [i.e., not an isomorphism class of functors]. Indeed, more generally, one could define a notion of “n-species” for arbitrary integers n ? 1. Since, however, this approach would only serve to add an unnecessary level of complexity to the theory, we choose here to take the approach of working with “functors considered up to isomorphism”. 0177現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/08(水) 19:04:57.09ID:1QCooAdl>>175 追加 species って、wikipedia では、下記 Combinatorial species なのだが、望月先生と同じ意味か? Andre Joyal 抜きには語れないようだが、望月 IUT4には Joyal先生の名前が出てこない(^^;
https://en.wikipedia.org/wiki/Combinatorial_species Combinatorial species (抜粋) In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures.
https://en.wikipedia.org/wiki/Andr%C3%A9_Joyal (抜粋) Andre Joyal (born 1943) is a professor of mathematics at the Universite du Quebec a Montreal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013,[1] where he was invited to join the Special Year on Univalent Foundations of Mathematics.[2]
Research He discovered Kripke?Joyal semantics,[3] the theory of combinatorial species and with Myles Tierney a generalization of the Galois theory of Alexander Grothendieck[4] in the setup of locales. Most of his research is in some way related to category theory, higher category theory and their applications. He did some work on quasi-categories, after their invention by Michael Boardman and Rainer Vogt, in particular conjecturing[5] and proving the existence of a Quillen model structure on sSet whose weak equivalences generalize both equivalence of categories and Kan equivalence of spaces. He co-authored the book "Algebraic Set Theory" with Ieke Moerdijk and recently started a web-based expositional project Joyal's CatLab [6] on categorical mathematics. 0178132人目の素数さん2020/01/08(水) 19:19:23.10ID:tPuJoa5y>>168 https://rio2016.5ch.net/test/read.cgi/math/1576852086/2840179132人目の素数さん2020/01/08(水) 19:26:07.17ID:LpZINTuE 知恵袋 https://chiebukuro.yahoo.co.jp/my/myspace_quedetail.php?writer=1043512917 https://chiebukuro.yahoo.co.jp/my/myspace_quedetail.php?writer=1147736549 二つのハンドルで質問しまくったがバカにされ始めたことを気づいたのか https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q13218505343 で ID を非公開にwwwwwwwwwwwww
バカはこんな簡単な英文も読めないらしいw For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right. 0203132人目の素数さん2020/01/10(金) 00:11:23.76ID:YnXkCflA>>195 高卒(工業高校)には選択公理も同値類もわからんわなぁ〜!!ww(^^; 0204132人目の素数さん2020/01/10(金) 00:24:01.02ID:YnXkCflA>>197 記事後半ははっきり言って無価値で無意味
そしてバカは訳も分からず不成立派の尻馬に乗ってるだけw バカ丸出しw 0205132人目の素数さん2020/01/10(金) 00:26:47.94ID:YnXkCflA>>200 >ここ、別に難しい話じゃない いや、時枝記事をまったく読めてないバカが短絡してるだけだからw バカ丸出しw 0206現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/10(金) 00:43:07.94ID:KeHo+Wgs>>202 (引用開始) For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right. (引用終り)
・Pruss氏のAnswerより(冒頭部分) The probabilistic reasoning depends on a conglomerability assumption, namely that given a fixed sequence u→ , the probability of guessing correctly is (n?1)/n, then for a randomly selected sequence, the probability of guessing correctly is (n?1)/n. But we have no reason to think the event of guessing correctly is measurable with respect to the probability measure induced by the random choice of sequence and index i, and we have no reason to think that the conglomerability assumption is appropriate.
・Our choice of index i is made randomly, but for this we only need the uniform distribution on {0,…,n}. It is made independently of the opponent's choice. ? Denis Dec 17 '13 at 15:21
・What we have then is this: For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n?1)/n. That's right. But now the question is whether we can translate this to a statement without the conditional "For each fixed opponent strategy". ? Alexander Pruss Dec 19 '13 at 15:05
・How about describing the riddle as this game, where we have to first explicit our strategy, then an opponent can choose any sequence. then it is obvious than our strategy cannot depend on the sequence. The riddle is "find how to win this game with proba (n-1)/n, for any n." ? Denis Dec 19 '13 at 19:43
・But the opponent can win by foreseeing what which value of i we're going to choose and which choice of representatives we'll make. I suppose we would ban foresight of i? ? Alexander Pruss Dec 19 '13 at 21:25 (引用終り)
これで、ここでのPruss氏の発言は終わっている で、Denis Dec 17 '13 at 15:21 の”we only need the uniform distribution on {0,…,n}”を受けて Pruss氏 ”we win with probability at least (n?1)/n. That's right. But・・”でしょ
つまり、Denis氏の”the uniform distribution on {0,…,n}”を仮定すれば、(n?1)/nだというのだが でも、それは、Pruss氏のAnswer(冒頭部分)にある通り、 ”The probabilistic reasoning depends on a conglomerability assumption”という文脈で語っているのであって (この冒頭部分での、”the probability of guessing correctly is (n?1)/n. But・・”と符合しているのだが) その後の、”But・・”の部分がPruss氏の主張ですよ(;p 以上 0209現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/10(金) 00:47:00.88ID:KeHo+Wgs>>207-208 文字化け訂正
>But the opponent can win by foreseeing what which value of i we're going to choose and which choice of representatives we'll make. I suppose we would ban foresight of i? ? Alexander Pruss Dec 19 '13 at 21:25 Prussは愚かにも「ランダム選択される i を予測することで勝てる」と言っているが、どうやったら予測できるのかについては華麗にスルーw 当たり前である。予測できたらランダムとは言わないw つまり But 以下はPrussの負け惜しみw
https://en.wikipedia.org/wiki/Alexander_Pruss Alexander Pruss (抜粋) Alexander Robert Pruss (born January 5, 1973) is a Canadian mathematician, philosopher, Professor of Philosophy and the Co-Director of Graduate Studies in Philosophy at Baylor University in Waco, Texas. Biography Pruss graduated from the University of Western Ontario in 1991 with a Bachelor of Science degree in Mathematics and Physics. After earning a Ph.D. in Mathematics at the University of British Columbia in 1996 and publishing several papers in Proceedings of the American Mathematical Society and other mathematical journals,[4] he began graduate work in philosophy at the University of Pittsburgh.
http://alexanderpruss.com/cv.html Curriculum Vitae Alexander R. Pruss December, 2018 (抜粋) Books Infinity, Causation and Paradox, Oxford University Press, 2018
https://books.google.co.jp/books?id=RXBoDwAAQBAJ&printsec=frontcover&dq=Infinity,+Causation+and+Paradox&hl=ja&sa=X&ved=0ahUKEwjtzezLzPfmAhW9JaYKHZNcBi0Q6AEILDAA#v=onepage&q=Infinity%2C%20Causation%20and%20Paradox&f=false Infinity, Causation, and Paradox 著者: Alexander R. Pruss 0216現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/10(金) 08:17:32.90ID:KeHo+Wgs>>214 確率パラドックスの記事だよ(^^; 0217132人目の素数さん2020/01/10(金) 09:49:24.99ID:YnXkCflA>>216 相変わらずバカ丸出し 時枝は確率の話ではない、選択公理・同値類の話 バカだからそれが分からないだけw
The Riddle: We assume there is an infinite sequence of boxes, numbered 0,1,2,…. Each box contains a real number. No hypothesis is made on how the real numbers are chosen. You are a team of 100 mathematicians, and the challenge is the following: each mathematician can open as many boxes as he wants, even infinitely many, but then he has to guess the content of a box he has not opened. Then all boxes are closed, and the next mathematician can play. There is no communication between mathematicians after the game has started, but they can agree on a strategy beforehand. You have to devise a strategy such that at most one mathematician fails. Axiom of choice is allowed.
(解法略)
The Modification: I would find the riddle even more puzzling if instead of 100 mathematicians, there was just one, who has to open the boxes he wants and then guess the content of a closed box. He can choose randomly a number i between 0 and 99, and play the role of mathematician number i. In fact, he can first choose any bound N instead of 100, and then play the game, with only probability 1/N to be wrong. 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. (引用終り) 以上 0223現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/10(金) 11:24:33.45ID:ebMXZTdz>>222 追加