https://arxiv.org/pdf/1212.5740.pdf Filters and Ultrafilters in Real Analysis 2012 Max Garcia Mathematics Department California Polytechnic State University
P16 3.2 Finite, Infinitesimal, and Infinitely Large Numbers
3.2.1 Definition (Classification). Let x ∈*R (a) x is infinitesimal if | x |< ε for all ε ∈ R+. We denote the set of all infinitesimals by I(*R). 0475粋蕎 ◆C2UdlLHDRI 2020/08/23(日) 20:26:35.95ID:EERKJb15 あ。完全に瀬田氏、冗談抜きの正気の本気で、どれが誰の主張か分かっとらんのか｡ 文学で分かる文章に書き直してやったのに未だに理解できんのか… もう瀬田氏は大学数学の前に小学から中学に至る迄位の国語を学習し直した方がええわ 0476現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/24(月) 07:32:12.38ID:+oiN9Lqm おっさん、スレ違い 連続体仮説、下記 20世紀前半まで、連続体仮説を巡って、喧々がくがくの議論があった 20世紀の後半になって、「連続体仮説は証明も反証もできない命題である」ということが明確に証明されてた いま、喧々がくがくの議論をする人はいない
This number is equal to 1. In other words, "0.999..." and "1" represent the same number. (In other systems, 0.999... can have the same meaning, a different definition, or be undefined.) More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons?such as rigorous proofs relying on non-elementary techniques, properties, or disciplines?some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.
Infinitesimals The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. 0479１３２人目の素数さん2020/08/24(月) 07:52:34.47ID:Z6P5UFQD>>478 >a)とb)と、両方あるんじゃねと、テレンスタオはいう（下記） まだ懲りてないのか？おまえはコピペ以外何も喋るな 0480粋蕎 ◆C2UdlLHDRI 2020/08/24(月) 07:58:04.63ID:UFbgwNy8 下手に連続体仮説を喩えに出しとるが、要するに瀬田氏は 準超実数 且つ 順序体 ( に就き自動的に 準超実数体 ) で 非超現実数 の時 「0.999≠1は証明できるとも反証(⇔0.999…=1の証明)できるとも言えない命題である」 と主張する訳じゃな？ そう主張するなら立場を明確にする意味で当レス鍵括弧を中身丸事、コピペしつつ正式に肯定して見せよ｡ 0481粋蕎 ◆C2UdlLHDRI 2020/08/24(月) 08:29:23.67ID:UFbgwNy8>>478 日本語も英語も読めんのか？言うたのはタオ本人じゃのうてイアンて書いてあるじゃろ、と何度、言わせる？ よく読めばイアンも自身が0.999…≠1と思うとる訳じゃのうて学生の弁護で0.999…≠1と書いとるだけで､ 其の詳細は0.999…;…000000…≠0.999…;…999000…≠0.999…;…999999…=1と親切に書かれとろうが､ そのコピペの元のWikipediaの本国版にも日本語版にも。何でそう自分に都合良い様に曲解読みするん？ イアンを貶しめたいんか？しかもイアンの発言なのにタオの発言である様に言って、タオも貶めたいんか？
補足資料下記 熟読下さい（＾＾ https://en.wikipedia.org/wiki/Infinitesimal Infinitesimal (抜粋) Infinitesimals in teaching Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.[14][15]
14. Ely, Robert (2010). "Nonstandard student conceptions about infinitesimals" (PDF). Journal for Research in Mathematics Education. 41 (2): 117?146. JSTOR 20720128. Archived (PDF) from the original on 2019-05-06. 15. Katz, Karin Usadi; Katz, Mikhail G. (2010). "When is .999... less than1?" (PDF). The Montana Mathematics Enthusiast. 7 (1): 3?30. arXiv:1007.3018. ISSN 1551-3440. Archived from the original (PDF) on 2012-12-07. Retrieved 2012-12-07.
Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Consequently present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it.[4]（訳: 今日では、解析学の授業において無限小量について述べることはあまり一般的ではない。その結果、当世の学生はこの言葉づかいに全く習熟していない。にも拘らず、未だにそれを扱うことが必要である）
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf Research Institute for Mathematical Sciences - Kyoto University, Japan PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元 Online Seminar - Algebraic & Arithmetic Geometry Laboratoire Paul Painleve - Universite de Lille, France Version 1 - ε? - 09/10/2020
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/IUT-references.html Promenade in Inter-Universal Teichmuller Theory Org.: Collas (RIMS); Debes, Fresse (Lille).
The Programme of the seminar contains a selection of ~30 references with respect to (1) Diophantine Geometry, (2) IUT Geometry, and (3) Anabelian Geometry. We indicate some links towards the key opuses as well as some complementary notes and proceedings. 0536現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/17(土) 10:25:45.84ID:02Kfs2KS メモ https://afst.centre-mersenne.org/item/?id=AFST_2009_6_18_S2_5_0 https://afst.centre-mersenne.org/article/AFST_2009_6_18_S2_5_0.pdf The Way to the Proof of Fermat’s Last Theorem Gerhard Frey Annales de la Faculte des sciences de Toulouse : Mathematiques, Serie 6, Tome 18 (2009) no. S2, pp. 5-23.
http://backup.itsoc.org/review/05pl1.pdf The Way to the Proof of Fermat ’s Last Theorem Gerhard Frey 1This paper is based on a talk at the ISIT meeting 1997. The author wants to thank the organizers for the invitation and the warm hospitality 0537ぷっちゃん2020/10/17(土) 12:01:32.25ID:QjI40yYH ＞よくわからない定義に出くわしたとき，
幕府クン（＝慶喜クン） 「わけわからない発言で煙に巻いて誤魔化す」
数学に興味ないくせに、わかった風な顔をしたがるペテン師の態度ですねｗ 0538現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/17(土) 16:31:00.47ID:02Kfs2KS 下記、Goldfeld, Modular forms, elliptic curves, and the ABC-conjecture が、なかなか良いね
§1. The ABC-Conjecture. The ABC-conjecture was first formulated by David Masser and Joseph Osterl´e (see [Ost]) in 1985. Curiously, although this conjecture could have been formulated in the last century, its discovery was based on modern research in the theory of function fields and elliptic curves, which suggests that it is a statement about ramification in arithmetic algebraic geometry. The ABC-conjecture seems connected with many diverse and well known problems in number theory and always seems to lie on the boundary of what is known and what is unknown. We hope to elucidate the beautiful connections between elliptic curves, modular forms and the ABC-conjecture. Conjecture (ABC). Let A, B, C be non-zero, pairwise relatively prime, rational integers satisfying A + B + C = 0. Define N = Πp|ABC p to be the squarefree part of ABC. Then for every ε > 0, there exists κ(ε) > 0 such that max(|A|, |B|, |C|) < κ(ε)N1+ε. A weaker version of the ABC-conjecture (with the same notation as above) may be given as follows. Conjecture (ABC) (weak). For every ε > 0, there exists κ(ε) > 0 such that |ABC| 1/3 < κ(ε)N1+ε.
Conjecture. (Szpiro, 1981) Let E be an elliptic curve over Q which is a global minimal model with discriminant Δ and conductor N. Then for every ε > 0, there exists κ(ε) > 0 such that Δ < κ(ε)N6+ε. We show that Szpiro’s conjecture above is equivalent to the weak ABC-conjecture. Let A, B, C be coprime integers satisfying A + B + C = 0 and ABC 6= 0. Set N = Πp|ABCp. Consider the Frey-Hellegouarch curve EA,B : y2 = x(x - A)(x + B). A minimal model for EA,B has discriminant (ABC)2・ 2-s and conductor N ・ 2-t for certain absolutely bounded integers s, t, (see Frey [F1]). Plugging this data into Szpiro’s conjecture immediately shows the equivalence.
[F1] FREY, G., Links between stable elliptic curves and certain diophantine equations, Annales Universiatis Saraviensis, Vol 1, No. 1 (1986), 1-39. [F2] FREY, G., Links between elliptic curves and solutions of A-B=C, Journal of the Indian Math. Soc. 51 (1987), 117-145. (引用終り) 以上 0540ぷっちゃん2020/10/17(土) 17:23:37.39ID:QjI40yYH>>538 モジュラー形式も楕円曲線も理解できないシロウトには無縁だね
江戸総攻撃の前に行なわれた勝と新政府軍参謀西郷隆盛との交渉により、 江戸城は4月11日に新政府軍に明け渡された。 彰義隊や旧幕臣の暴発を恐れた慶喜は 4月11日午前3時に寛永寺大慈院を出て水戸へ向かった。 水戸では弘道館の至善堂にて引き続き謹慎した後、 7月に徳川家が駿府に移封されると、慶喜も駿河の宝台院に移って謹慎した。 これにより、徳川家による政権は幕を閉じた。 0546１３２人目の素数さん2020/10/18(日) 14:26:59.10ID:ufbJ1e15 フロべニオイドって自然な定義なのか？ 0547現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日) 19:38:34.42ID:ZLSkSSTT メモ貼る https://stacks.math.columbia.edu/bibliography The Stacks project Table of contentsBibliography (抜粋) Grothendieck, A., Standard conjectures on algebraic cycles Grothendieck, Alexander, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA 2) Grothendieck, Alexander, Fondements de la geometrie algebrique Grothendieck, Alexander, La theorie des classes de Chern Grothendieck, Alexander, Revetements etales et groupe fondamental (SGA 1) Grothendieck, Alexander, Sur quelques points d'algebre homologique Grothendieck, Alexander, Technique de descente et theoremes d'existence en geometrie algebrique. I. Generalites. Descente par morphismes fidelement plats Grothendieck, Alexander, Technique de descente et theoremes d'existence en geometrie algebrique. II. Le theoreme d'existence en theorie formelle des modules Grothendieck, Alexander, Techniques de construction et theoremes d'existence en geometrie algebrique. III. Preschemas quotients Grothendieck, Alexander, Techniques de construction et theoremes d'existence en geometrie algebrique. IV. Les schemas de Hilbert Grothendieck, Alexander and Dieudonne, Jean, Elements de geometrie algebrique I Grothendieck, Alexander and Dieudonne, Jean, Elements de geometrie algebrique I Grothendieck, Alexander and Dieudonne, Jean, Elements de geometrie algebrique II Grothendieck, Alexander and Murre, Jacob P., The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme Grothendieck, Alexander and Raynaud, Michel and Rim, Dock Sang, Groupes de monodromie en geometrie algebrique. I Grothendieck, Alexandre, Seminaire de geometrie algebrique du Bois-Marie 1965-66, Cohomologie l-adique et fonctions L, SGA5 0548現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日) 19:51:52.82ID:ZLSkSSTT>>546 >フロべニオイドって自然な定義なのか？
P3 Modus Operandi & Leitfaden. As a new geometry, the essence of Mochizuki’s IUT is to introduce a new semiotic system - formalism, terminology, and their interactions - that can be unsettling at first. This programme proposes a 3 layers approach with precise references, examples, and analogies. Because IUT discovery also benefits from a non-linear and spiralling approach, we provide further indications for an independent wandering: Mochizuki recommends to start with the introductory [Alien] - young arithmetic-geometers can also consult [Fes15] for a shorter overview. We also recommend to begin with §Intro - §3.6-7 ibid. for a direct encounter with IUT’s semiotic, then to follow one’s own topics of interest according to Fig. 1, which also indicates some topic-wise references as entry-points - [EtTh], [GenEll], etc. Within the “canon” [IUTChI]-[IUTChIV], our recommendation is to start with [IUTChIII] §Introduction. Intuition of the reader can further rely on the strongly consistent terminology of IUT - e.g. Frobenioid, mono-anabelian transport, arithmetic analytic.
※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.
※ Hodge-Arakelov and p-adic Teichmuller theories stand as important models for IUT, which also relies on key categorical constructions - e.g. Frobenioids and anabelioids. These aspects are not included in this programme - we refer to [Alien] and the canon for references - they can be the object of additional talks by specialists. 以上 0554現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/20(火) 08:06:28.04ID:V6fYxSC9https://rio2016.5ch.net/test/read.cgi/math/1600350445/541 IUT と ABC予想 (応援スレ) 49
>※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元 Online Seminar - Algebraic & Arithmetic Geometry Laboratoire Paul Painleve - Universite de Lille, France (抜粋) P2 ※ In order to keep the length of this guide (incl. 〜 25 tables, figures, and diagrams) strictly shorter than the IUT corpus - 〜 1200 pages with a piece of anabelian geometry, 〜 675 pages for the canon, and 〜 170 pages for the introductory [Alien] - some details have been omitted, some approximations were made; they should be negligible for our goal. Content will be updated according to the progress of the seminar, see version and date.
P3 Within the “canon” [IUTChI]-[IUTChIV], our recommendation is to start with [IUTChIII] §Introduction. Intuition of the reader can further rely on the strongly consistent terminology of IUT - e.g. Frobenioid, mono-anabelian transport, arithmetic analytic.
※ Hodge-Arakelov and p-adic Teichmuller theories stand as important models for IUT, which also relies on key categorical constructions - e.g. Frobenioids and anabelioids. These aspects are not included in this programme - we refer to [Alien] and the canon for references - they can be the object of additional talks by specialists.
つづく 0555現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/20(火) 08:07:05.29ID:V6fYxSC9>>554 つづき >※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.
そうか、この[Alien]っていうのが、重要な論文なんだね〜（＾＾ P4 [Alien]： [Alien] S. Mochizuki, “The mathematics of mutually alien copies: From Gaussian integrals to Inter-universal Teichmuller theory,” RIMS Preprint no. 1854, 169p. Jul. 2016, Eprint available on-line.
http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf [7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-04-04)
Abstract Inter-universal Teichm¨uller theory may be described as a construction of certain canonical deformations of the ring structure of a number field equipped with certain auxiliary data, which includes an elliptic curve over the number field and a prime number ? 5. In the present paper, we survey this theory by focusing on the rich analogies between this theory and the classical computation of the Gaussian integral. The main common features that underlie these analogies may be summarized as follows: ・ the introduction of two mutually alien copies of the object of interest; ・ the computation of the effect -i.e., on the two mutually alien copies of the object of interest -of two-dimensional changes of coordinates by considering the effect on infinitesimals;
・ the passage from planar cartesian to polar coordinates and the resulting splitting, or decoupling, into radial -i.e., in more abstract valuation-theoretic terminology, “value group” -and angular -i.e., in more abstract valuation-theoretic terminology, “unit group” -portions; ・ the straightforward evaluation of the radial portion by applying the quadraticity of the exponent of the Gaussian distribution; ・ the straightforward evaluation of the angular portion by considering the metric geometry of the group of units determined by a suitable version of the natural logarithm function.
[Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”.] After reviewing the classical computation of the Gaussian integral, we give a detailed survey of inter-universal Teichm¨uller theory by concentrating on the common features listed above. The paper concludes with a discussion of various historical aspects of the mathematics that appears in inter-universal Teichm¨uller theory. (引用終り) 以上 0557１３２人目の素数さん2020/10/20(火) 16:18:17.10ID:8nlx/Wj4>>552 ＞頭に入れておくのが良い
https://en.wikipedia.org/wiki/Brian_Conrad Brian Conrad Brian Conrad (born November 20, 1970), is an American mathematician and number theorist, working at Stanford University. Previously, he taught at the University of Michigan and at Columbia University. Conrad and others proved the modularity theorem, also known as the Taniyama-Shimura Conjecture. He proved this in 1999 with Christophe Breuil, Fred Diamond and Richard Taylor, while holding a joint postdoctoral position at Harvard University and the Institute for Advanced Study in Princeton, New Jersey.
https://en.wikipedia.org/wiki/Kiran_Kedlaya Kiran Sridhara Kedlaya (/?k?r?n ??ri?d?r k?d?l??j?/;[2] born July 1974) is an Indian American mathematician. He currently is a Professor of Mathematics and the Stefan E. Warschawski Chair in Mathematics[3] at the University of California, San Diego. 0559現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/20(火) 17:34:00.99ID:lsCoo7pb>>558 誤変換タイポ訂正
