(過去スレより ご参考) http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf [4] Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations. PDF NEW !! (2019-10-31)
P3 Theorem A. (Diophantine Inequalities) Then, relative to the notation of [GenEll] [reviewed in the discussion preceding Corollary 2.2 of the present paper], one has an inequality of “bounded discrepancy classes” Thus, Theorem A asserts an inequality concerning the canonical height [i.e., “htωX(D)”], the logarithmic different [i.e., “log-diffX”], and the logarithmic conductor [i.e., “log-condD”] of points of the curve UX valued in number fields whose extension degree over Q is <= d . In particular, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A. We refer to [Vjt] for a detailed exposition of these conjecture 0100現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/05(日) 16:41:12.28ID:dWKXmW0r>>90 追加ご参考 (芽・茎と、同値類) https://ja.wikipedia.org/wiki/%E8%8A%BD_(%E6%95%B0%E5%AD%A6) 芽 (数学) (抜粋) 名前は層 (sheaf) のメタファーの続きで cereal germ に由来している。穀物にとってそうであるように芽は(局所的に)関数の「心臓 (heart)」であるからだ。
正式な定義 基本的な定義 x で同じ芽を定義することが(写像や集合の上で)同値関係であることを確かめることは直截であり、その同値類を芽(それぞれ写像の芽あるいは集合の芽)と呼ぶ。同値関係は通常 f〜 x gあるいは S〜x T と書かれる。
基本的な性質 f と g が x において同値な芽であれば、それらは連続性や微分可能性といったすべての局所てな性質を共有し、したがって可微分あるいは解析的芽などについて話すことは意味をなす: 部分集合に対しても同様である。芽の1つの代表が解析的集合であれば、すべての代表は少なくとも x のある近傍上で解析的である。 同値類は前層 F の x における茎(英語版) Fxをなす。この同値関係は上で記述された芽同値の抽象化である。
https://ja.wikipedia.org/wiki/%E8%8C%8E_(%E6%95%B0%E5%AD%A6) 茎 (数学) (抜粋) 直極限の定義(あるいは普遍性)により,茎の元は元 x_{U}∈ {F}(U)の同値類である,ただし2つのそのような切断 xU と xV は2つの切断の制限が x のある近傍上で一致するときに同値であると考える.
注意 x を含む任意の開集合 U に対して自然な射 F(U) → Fx が存在する:それは F(U) における切断 s をその芽 (germ), すなわち直極限におけるその同値類に送る.これは芽の通常の概念の一般化であり,X 上の連続関数の層の茎を見ることで復元できる. 0101132人目の素数さん2020/01/05(日) 16:41:45.41ID:WDR2q7oi それじゃ、おっちゃんもう寝る。 0102132人目の素数さん2020/01/05(日) 16:53:37.85ID:WDR2q7oi>>90 >>100 そうそう、確率の話だから最初はn次元 Euclid とかで実解析的に考えるモンだ。 いきなり複素平面Cやn次元複素数空間 C^n とかで考えても意味ない。
あと、下記が参考になる (なぜ、mathoverflow>>465 の手法が成立たないのか? ”CONGLOMERABILITY”が成立ってないというのが、数学DR Alexander Pruss氏の指摘(2013)で、それを2018年の著書で詳しく解説している) スレ65 https://rio2016.5ch.net/test/read.cgi/math/1557142618/750-754 https://books.google.co.jp/books?id=RXBoDwAAQBAJ&printsec=frontcover&dq=Infinity+Paradox+Pruss+2018&hl=ja&sa=X&ved=0ahUKEwiQ8tDxr-zmAhVdy4sBHd5cAlkQ6AEIKTAA#v=onepage&q=Infinity%20Paradox%20Pruss%202018&f=false Infinity, Causation, and Paradox 著者: Alexander R. Pruss Oxford University Press, 2018 P75 (抜粋) 2.5.3 COUNTABLE ADDITITVITY AND CONGLOMERABILITY (引用終り)
(mathoverflowの”conglomerability”関連箇所) https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice Probabilities in a riddle involving axiom of choice Dec 9 '13 (抜粋) (Alexander Pruss氏) <12> (抜粋) The probabilistic reasoning depends on a conglomerability assumption・・ 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. A quick way to see that the conglomerability assumption is going to be dubious is to consider the analogy of the Brown-Freiling argument against the Continuum Hypothesis (see here for a discussion). http://www.mdpi.com/2073-8994/3/3/6360110132人目の素数さん2020/01/05(日) 20:51:27.29ID:p9Somwtl>>108 >貴方はレベル高そう >多分私よりも 見苦しいな 時枝不成立派とみるや途端におべっかw まあそう慌てなさんな、最初は誰しも不成立に見える問題だから(そこが数学パズルたる所以)
>”あーあーあの話”というのは、一様分布の区間が、有限と無限とで異なるって話かな? >非正則な分布の話は、過去スレにもあるが バカだねえ〜 箱の中身の分布は指定されていないと彼は言ってるんだよw 時枝記事で指定されている分布はただ一つ 「さて, 1〜100 のいずれかをランダムに選ぶ. 」 すなわち列の選択が一様分布、それだけw おまえホントに何にも分かってないなw バカ丸出しw 0111132人目の素数さん2020/01/05(日) 20:55:43.20ID:p9Somwtl>>109 >貴方のために、下記のDr Pruss氏 を バカ丸出しw Prussは勝率99/100以上を認めているw 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.
7章”The Topological Setting”とかなっていて P9 ”In Chapter 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. The most natural setting for this is a topological space and if we wanted to only consider continuous colorings, then the limit operator would serve as a unique optimal predictor. But we want to consider arbitrary colorings. Thus we have each point in a topological space representing an agent and if f and g are two colorings, then f ≡a g if f and g agree on some deleted neighborhood of the point a. It turns out that an optimal predictor in this case is wrong only on a set that is "scattered" (a concept with origins going back to Cantor). Moreover, this predictor again turns out to be essentially unique, and this is the main result in Chapter 8.” などとある さすれば、時枝もそのままじゃ(Topologicalな条件を加えないと)、成り立たないと思う (引用終り) 以上 0116現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2020/01/06(月) 07:30:42.33ID:t3ENnC2G>>113 (引用開始) Prussの >random choice of sequence は明らかに誤り。 なぜなら時枝ゲームのルールでは数列(無限個の箱の中身)の決め方は出題者の任意であり、ランダムではない。 「箱がたくさん,可算無限個ある.箱それぞれに,私が実数を入れる. どんな実数を入れるかはまったく自由」 Prussはこの時点で間違っているので、その後の確率測度の非可測性にもとづく主張はまったく無意味。 (引用終り)
Gくん、がんばって ・3.12について、解説100ページくらい書いて、パワーポイントも別に作って、ワークショップで議論してほしいな 得に、”「大元誤解」の本質”の謎解きと解説をしっかり ・あと、IUTその4の P74 Remark 3.3.1. (i) One well-known consequence of the axiom of foundation of axiomatic set theory is the assertion that “∈-loops” の話も、ちゃんと解説してほしいね ・同 P68 Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that it may be possible to justify the stance of ignoring such issues in the context of the present series of papers なんて話も、もうちょっとすっきりさせてほしいね 0140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) 00:27:46.48ID:b2sufDWR>>16 > https://ja.yourpedia.org/wiki/%E5%AE%87%E5%AE%99%E9%9A%9B%E3%82%BF%E3%82%A4%E3%83%92%E3%83%9F%E3%83%A5%E3%83%A9%E3%83%BC%E7%90%86%E8%AB%96 > 宇宙際タイヒミュラー理論 Yourpedia
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/1576852086/284