Lindenstrauss has made far-reaching advances in ergodic theory, the study of measure preserving transformations. His work on a conjecture of Furstenberg and Margulis concerning the measure rigidity of higher rank diagonal actions in homogeneous spaces has led to striking applications. Specifically, jointly with Einsiedler and Katok, he established the conjecture under a further hypothesis of positive entropy. It has impressive applications to the classical Littlewood Conjecture in the theory of diophantine approximation. Developing these as well other powerful ergodic theoretic and arithmetical ideas, Lindenstrauss resolved the arithmetic quantum unique ergodicity conjecture of Rudnick and Sarnak in the theory of modular forms. He and his collaborators have found many other unexpected applications of these ergodic theoretic techniques in problems in classical number theory. His work is exceptionally deep and its impact goes far beyond ergodic theory.
これが、フィールズ賞論文の一つみたいだね http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n1-p03.pdf Invariant measures and arithmetic quantum unique ergodicity By Elon Lindenstrauss* Appendix with D. Rudolph Annals of Mathematics, 163 (2006), 165?219 (抜粋) Abstract We classify measures on the locally homogeneous space Γ\ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproof of the main result. (引用終り) 0128現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/24(水) 23:20:12.11ID:64Y6nh+a>>127 関連
"Littlewood’s conjecture"への言及があるね(^^
P170 "The scope of the methods developed in this paper is substantially wider than what I discuss here. In particular, in a forthcoming paper with M. Einsiedler and A. Katok [EKL06] we show how using the methods developed in this paper in conjunction with the methods of [EK03] one can substantially sharpen the results of the latter paper. These stronger results imply in particular that the set of exceptions to Littlewood’s conjecture, i.e. those (α, β) ∈ R2 for which lim n→∞ n||nα|| ||nβ|| > 0, has Hausdorff dimension 0." 0129現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/25(木) 10:53:47.49ID:v5l3CFRR>>111 関連
実際は、Lindenstrauss のフィールズ賞に直結した論文は、Annals of Math (2006)に、二編に分けて投稿され 量子エルゴードは第一論文であり、"Littlewood’s conjecture"は次の論文で”with M. Einsiedler and A. Katok [EKL06] ”だったわけだが、 第二論文("Littlewood’s conjecture")について紹介している和文PDFなどは、ほとんど検索でヒットしなかった
賛成だな。実際、例えば、>>65のThe Straddle Lemmaでは ”Lemma 4.3. Straddle Lemma. Let F : I → R be differentiable at a point t ∈ I. Given ε there exists ε(t) > 0 such that if u, v ∈ I satisfy t − δε(t) <= u <= t <= v <= t + δε(t) (4.3) then we have |F(v) − F(u) − F'(t)(u − v)| <= ε(v − u) (4.4)” のようにして、ε-δに対して、∀や∃とかの論理記号は使っていないし 和文のテキスト(教科書)でも、∀や∃を使う方が少ないと思っている
一般に、任意のヘンストック=クルツヴァイル可積分函数はルベ−グ可測であり、また f がルベ−グ可積分であるための必要十分条件は f および|f|がともにヘンストック=クルツヴァイル可積分となることである。 これは、ヘンストック=クルツヴァイル積分を、「非絶対可積分」版ルベ−グ積分と看做すことができることを意味する。 またこれから、ヘンストック=クルツヴァイル積分が単調収束定理の適当な(函数が非負であることを課さない)変形版を満たすことや、 優収斂定理の適当な変形版(函数列 fn に対する支配条件を弱めて、適当な可積分函数 g, h で g ≤ fn ≤ h とできるとしたもの)を持たすことが導かれる。
函数 F が至る所(若しくは可算個の例外を除く至る所)微分可能ならば、導函数 F′ はヘンストック=クルツヴァイル可積分で、その不定ヘンストック=クルツヴァイル積分は F に一致する(F′ がルベ−グ可積分である必要はないことに注意)。すなわち、任意の可微分函数はその導函数の積分と定数の違いを除いて一致するという微分積分学の第二基本定理
F(x)−F(a)=∫a〜x F'(t)dt.
がより簡潔でより十分な形で得られたことになる。逆に、ルベ−グの微分定理はヘンストック=クルツヴァイル積分に関しても成立する。すなわち、 f が [a, b] 上でヘンストック=クルツヴァイル可積分で
F(x)=∫a〜x f(t)dt
を満たすならば、[a, b] の殆ど至る所で F′(x) = f(x) が成立する(特に F は殆ど至る所微分可能である)。
>>137 >クザンの定理(英語版)によれば、どのようなゲ−ジ δ に対してもこのような δ−細分割 P は存在する。 >したがって、この条件は空虚な真理(Vacuous truth、この場合どのようなゲ−ジ δ を選んでも δ−細分割である P が存在しないために上記の条件が真になること)とはなり得ない。
原文: Cousin's theorem states that for every gauge δ, such a δ-fine partition P does exist, so this condition cannot be satisfied vacuously. (google訳の微修正) Cousinの定理によれば、すべてのゲージδについて、そのようなδファイン・パーティションPが存在するので、この条件は空にならない。
>>138 >またこれから、ヘンストック=クルツヴァイル積分が単調収束定理の適当な(函数が非負であることを課さない)変形版を満たすことや、 >優収斂定理の適当な変形版(函数列 fn に対する支配条件を弱めて、適当な可積分函数 g, h で g ≤ fn ≤ h とできるとしたもの)を持たすことが導かれる。
原文: It also implies that the Henstock-Kurzweil integral satisfies appropriate versions of the monotone convergence theorem (without requiring the functions to be nonnegative) and dominated convergence theorem (where the condition of dominance is loosened to g(x) <= fn(x) <= h(x) for some integrable g, h).
とゲ−ジと呼ばれる正値函数 δ: [a, b] → (0, ∞) に対して、点付き分割 P が δ−細 (δ−fine) であるとは、各 i について
t_i−δ(t_i) < u_i−1 <= t_i <= u_i<t_i+δ (t_i)
を満たすことである。」 (引用終わり)
この定義、まさに、Straddle Lemma を使用? かな(^^
(参考) スレ49 https://rio2016.5ch.net/test/read.cgi/math/1514376850/186 より (抜粋) 補題(straddle lemma) f : R → R は点x ∈ R で微分可能とする. このとき, 次が成り立つ. ∀ε > 0, ∃δ > 0, ∀y, z ∈ R [ x − δ <= y <= x <= z <= x + δ)→ |f(z) − f(y) − f’(x)(z − y)| <= ε(z − y) ] . この補題がstraddle (またぐ・またがる) と呼ばれているのは, y とz を「x をまたぐように取る」 からである. そして, (*) の計算は, この補題の証明と同じ考え方を適用したに過ぎない. 結局, 全体としては, 極めてオーソドックスかつ簡単な議論で定理1.7 が証明できたことになる. QED (引用終わり) 0151現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/25(木) 18:10:13.88ID:v5l3CFRR>>149 おう、ありがとう あなたのオナラのおかげで、スレの勢い35まで上がったぜ!(^^ 0152132人目の素数さん2018/01/25(木) 21:19:34.58ID:5xFBb0e5 >私としては、和文を読んでみたかったのだが・・(^^ >(私の英語レベルでは、圧倒的に読むスピードが和文の方が早いので・・(^^ ) 一年生用教科書を読めない君には和文も英文も無い 0153132人目の素数さん2018/01/25(木) 21:23:00.21ID:5xFBb0e5 >∀や∃とかの論理記号は、余程論理が複雑になったりするようなことや何らかの事情がない限り、むしろ使わない方が望ましい。 ∀や∃を使わないでまともな証明を書こうと思ったらよっぽど複雑になる ごく普通に使われている記号を使わない理由は全く無い。 0154現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/25(木) 21:42:33.34ID:dgxCIZKr>>152-153 だから、おっちゃん、 自分のことを、人に当てはめるなよ(^^ 0155現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/25(木) 22:13:22.50ID:dgxCIZKr>>152-153 だから、おっちゃん、例えば・・ 米国の大人の論文のε-δ の表現は下記で、これ普通だろ(^^ ∀や∃を多用するのは、日本人のガキだけだろ
(>>82) http://classicalrealanalysis.info/documents/2323311.pdf THE TEACHING OF MATHEMATICS EDITED BY JOAN P. HUTCHINSON AND STAN WAGON More on the Fundamental Theorem of Calculus CHARLES SWARTZ Department of Mathematics, New Mexico State University, BRIAN S. THOMSON Department of Mathematics, Simon Fraser University, Burnaby, B. C., Canada The American Mathematical Monthly, Vol. 95, No. 7 (Aug. - Sep., 1988) (抜粋)
DEFINITION 1. A functionf : [a, b] → R is Riemann integrable over [a, b] if there exists A ∈ R such that for every ε > 0 there exists δ > 0 such that if P is a partition of mesh less than δ and if ti ∈ [xi-1, xi], then
| Σ i=1〜n f(ti)(xi - xi-1) - A | < ε.
The number A is called the Riemann integral of f and is denoted by ∫a〜b f.
(>>82 より) http://classicalrealanalysis.info/documents/2323311.pdf THE TEACHING OF MATHEMATICS EDITED BY JOAN P. HUTCHINSON AND STAN WAGON More on the Fundamental Theorem of Calculus CHARLES SWARTZ Department of Mathematics, New Mexico State University, BRIAN S. THOMSON Department of Mathematics, Simon Fraser University, Burnaby, B. C., Canada The American Mathematical Monthly, Vol. 95, No. 7 (Aug. - Sep., 1988) (抜粋) LEMMA 3 (STRADDLE LEMMA). Let F: [a, b] - R be differentiable at z ∈ [a, b]. Then for each ε > 0, there is a δ > 0 such that |F(v) - F(u) - F'(z)(v - u) | < ε(v -u), whenever u < z < v and [u, v] ⊆ [a, b] ∩ (z - δ, z + δ). (引用終り)
In calculus, an antiderivative, primitive function, primitive integral or indefinite integral[Note 1] of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F ′ = f.[1][2] (引用終わり) 0174現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/26(金) 15:55:41.64ID:wuOe0QB3 ”the straddle lemma”は、病的な関数には適用不可ということを、ご注意申し上げておく(^^ 0175現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/26(金) 15:56:25.82ID:wuOe0QB3>>172 おっちゃん、どうも、スレ主です。 おっちゃん、よく勉強しているね(^^ 0176132人目の素数さん2018/01/26(金) 16:01:35.93ID:IHQggStd>>175 ∀や∃、s.t. とかの記号が必要になるのは。 色々な記号を多用するような段階の話で、もっと後。 0177132人目の素数さん2018/01/26(金) 21:04:11.59ID:1kwwEpu6 ∀、∃なんて大学生以上には空気のようなもの、いちいち騒ぎ立てなさんな しかしスレ主のレベルの低さにはほとほと呆れかえる 0178現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/26(金) 21:26:16.08ID:GL3Hhw9j>>177
ご苦労さん(^^
(>>169より) http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Herschlag.pdf Greg Herschlag: A brief introduction to gauge integration (pdf)
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Herschlag.pdf Greg Herschlag: A brief introduction to gauge integration (pdf) 0182現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 09:35:59.44ID:pLEQonr1>>65 参考 > Before continuing it is important to acknowledge and note that all of the ideas of this paper are drawn from Robert G. Bartle’s A Modern Theory of Integration.
https://www.amazon.co.jp/Modern-Integration-Graduate-Studies-Mathematics/dp/0821808451 A Modern Theory of Integration (Graduate Studies in Mathematics) (英語) ハードカバー ? 2001/4/1 Robert G. Bartle (著) (抜粋) 内容紹介 The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is "better" because it removes some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with "improper" integrals. This book is an introduction to a relatively new theory of the integral (called the "generalized Riemann integral" or the "Henstock-Kurzweil integral") that corrects the defects in the classical Riemann theory and both simplifies and extends the Lebesgue theory of integration. Although this integral includes that of Lebesgue, its definition is very close to the Riemann integral that is familiar to students from calculus. One virtue of the new approach is that no measure theory and virtually no topology is required. Indeed, the book includes a study of measure theory as an application of the integral.
Part 1 fully develops the theory of the integral of functions defined on a compact interval. This restriction on the domain is not necessary, but it is the case of most interest and does not exhibit some of the technical problems that can impede the reader's understanding.
Part 2 shows how this theory extends to functions defined on the whole real line. The theory of Lebesgue measure from the integral is then developed, and the author makes a connection with some of the traditional approaches to the Lebesgue integral.
Thus, readers are given full exposure to the main classical results. The text is suitable for a first-year graduate course, although much of it can be readily mastered by advanced undergraduate students. Included are many examples and a very rich collection of exercises. There are partial solutions to approximately one-third of the exercises.
Excellent introduction to the theory of gauge integrals. A mild acquaintance with riemann integrals is all that's required to get started with this book. 8人のお客様がこれが役に立ったと考えています. (引用終り)
https://en.wikipedia.org/wiki/Robert_G._Bartle Robert G. Bartle (抜粋) Robert Gardner Bartle (1927 ? 2003) was an American mathematician specializing in real analysis. He is known for writing the popular textbooks The Elements of Real Analysis (1964), The Elements of Integration (1966), and Introduction to Real Analysis (2011) published by John Wiley & Sons.
Bartle was born in Kansas City, Missouri, and was the son of Glenn G. Bartle and Wanda M. Bartle. He was married to Doris Sponenberg Bartle (born 1927) from 1952 to 1982 and they had two sons, James A. Bartle (born 1955) and John R. Bartle (born 1958). He was on the faculty of the Department of Mathematics at the University of Illinois from 1955 to 1990.
Bartle was Executive Editor of Mathematical Reviews from 1976 to 1978 and from 1986 to 1990. From 1990 to 1999 he taught at Eastern Michigan University. In 1997, he earned a writing award from the Mathematical Association of America for his paper "Return to the Riemann Integral".[1]
References 1.^Jump up ^ Bartle, Robert G. (1996). "Return to the Riemann Integral". The American Mathematical Monthly. Mathematical Association of America. 103 (8): 625?632. doi:10.2307/2974874. JSTOR 2974874. ・Robert G. Bartle (1990) "A brief history of the mathematical literature". ・Jane E. Kister & Donald R. Sherbert (2004) "Robert G. Bartle (1927 ? 2003)". Notices of the American Mathematical Society 51(2):239?40. ・Robert G. Bartle, 75, Mathematician and Author, New York Times (2003) [1] (引用終り) 0186現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 10:04:20.89ID:pLEQonr1>>185 関連 > 1.^Jump up ^ Bartle, Robert G. (1996). "Return to the Riemann Integral". The American Mathematical Monthly. Mathematical Association of America. 103 (8): 625?632. doi:10.2307/2974874. JSTOR 2974874.
From: Several authors of more advanced books and articles --
Robert Bartle, USA <mth bartle@emuvax.emich.edu> Ralph Henstock, Ireland <r.henstock@ulst.ac.uk> Jaroslav Kurzweil, Czech Republic <kurzweil@mbox.cesnet.cz> Eric Schechter, USA <schectex@math.vanderbilt.edu> Stefan Schwabik, Czech Republic <schwabik@beba.cesnet.cz> Rudolf Vyborny, Australia <R.Vyborny@mailbox.uq.edu.au>
Subject: Replacing the Riemann integral with the gauge integral
In summary, we feel that the changes in the calculus book would not be major but would improve the teaching of calculus. We invite your questions on these matters.
Sincerely,
Robert Bartle, Ralph Henstock, Jaroslav Kurzweil, Eric Schechter, Stefan Schwabik, and Rudolf Vyborny (This letter is being distributed to publishers' representatives at the Joint Mathematics Meetings in San Diego, California, in January 1997. We may also try to publish it and/or distribute it in some other fashion in the near future, perhaps with slight modifications and/or more signatures.) (引用終り) 以上 0192現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 13:57:58.71ID:pLEQonr1 さて 問題の定理がなんのなのか? 新参読者には分らないだろうから、下記を貼る(^^
えーと、それで(上記178-186より要点引用) (抜粋) 補題1.5(注:”The Straddle Lemma”の変形) f : R → R とx ∈ R は lim sup y→x |(f(y) − f(x))/(y − x)|< +∞ を満たすとする. このとき, ある正整数N,M >= 1 に対して ∀y, z ∈ R [x − 1/M < y < x < z < x +1/M → |f(z) − f(y)| <= N(z − y)]が成り立つ.
定理1.7 (422 に書いた定理) f : R → R とする. Bf :={x ∈ R | lim sup y→x |(f(y) − f(x))/(y − x)|< +∞ } と置く: もしR−Bf が内点を持たない閉集合の高々可算和で被覆できるならば、 f はある開区間の上でリプシッツ連続である. 証明 このとき, 補題1.5 を満たすN,M >= 1 が存在するので, 明らかにx ∈ BN,M である.
系1.8 有理数の点で不連続, 無理数の点で微分可能となるf : R → R は存在しない. 証明 定理1.7 が使えて, f はある開区間(a, b) の上でリプシッツ連続である. 一方で, x ∈ Q とf の仮定により, f は点x で不連続である. これは矛盾. よって, 題意が成り立つ. (引用終り)
5)系1.8で、「有理数の点で不連続, 無理数の点で微分可能となるf : R → R は存在しない」ことを示すために、 定理1.7を適用して、”f はある開区間(a, b) の上でリプシッツ連続である”としているが、 系1.8の関数 f は、「病的な不連続(discontinuity) 点を持つ関数」であるから 定理1.7を適用するのは不適切であり、矛盾が導かれるとする背理法は不成立(それは、もともと適用ルール違反であり、矛盾が導かれるのは当然)
以上 0197現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 14:27:30.84ID:pLEQonr1 スレ49 https://rio2016.5ch.net/test/read.cgi/math/1514376850/41 (抜粋) REMARK BY RENFRO: The last theorem follows from the following stronger and more general result. Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set). This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function", Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below. (引用終り)
<参考> https://en.wikipedia.org/wiki/Sengupta Sengupta (抜粋) Sengupta is a surname found among Bengali people of India and Bangladesh. They belong to the Baidya caste.
Contents [hide] 1 Geographical distribution 2 Notables 3 References 4 See also Geographical distribution[edit] As of 2014, 67.8% of all known bearers of the surname Sengupta were residents of India and 22.5% were residents of Bangladesh. In India, the frequency of the surname was higher than national average in the following states and union territories:[1]
Ushoshi Sengupta, (born 1988) Indian beauty pageant contestant and winner of Miss India Universe 2010 (引用終り) 0198現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 14:31:05.91ID:pLEQonr1>>197 参考
http://www.afpbb.com/articles/modepress/2748221?pid=6076546 ミス・ユニバース出場者、本番に向けてリハーサル 2010年08月14日 12:28 発信地:ラスベガス/米国 AFPBB News 0199現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 14:52:03.60ID:pLEQonr1 ところで CHRISTOPHER P. CHAMBERS先生は、数学の専門家ではないが、下記”Intergenerational Equity: Sup, Inf, Lim Sup, and Lim Inf” ”6. Open problems ”が、どこまで正確か不明ですが http://chambers.georgetown.domains/ CHRISTOPHER P. CHAMBERS Professor of Economics Department of Economics Georgetown University
CV http://chambers.georgetown.domains/CPCVita.pdf (抜粋) B.S., Mathematics and Economics, with honors in mathematics, May 1998. University of Maryland, College Park. M.A., Economics, June 2001, University of Rochester. Ph.D., Economics, June 2003, University of Rochester. Supervisor: William Thomson. (引用終り)
Abstract We study the problem of intergenerational equity for utility streams and a countable set of agents. A numerical social welfare function is invariant to ordinal transformation, satis?es a weak monotonicity condition, and an invariance with respect to concatenation of utility streams if and only if it is either the sup, inf, lim sup, or lim inf. Keywords: intergenerational equity, supremum, limit superior.
6. Open problems An interesting fact is that these results do not extend to continuum of agents (or higher cardinalities of agents) models. For example, consider the rule, which is a kind of ?countable lim sup,?which speci?es that the utility of a society is the smallest value for which an at most countable number of agents receive at least that value. This rule satis?es all of the axioms we have posited, yet it is not strictly speaking a inf, sup, lim inf, or lim sup. An interesting question is to study these generalized limit concepts for arbitrary cardinalities of agents. (引用終り) 以上 0200現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 15:15:13.84ID:pLEQonr1 あんまり関係ないけど、ヒットしたので貼る(^^ http://www.math.nus.edu.sg/~matwujie/NUS-12-2006.pdf Topology and Poincare Conjecture 2006/12/14 Jie Wu Department of Mathematics National University of Singapore 0201現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/27(土) 18:12:04.34ID:pLEQonr1 栃ノ心優勝か。「ジョージア」は、昔は”グルジア”とか言ったかも(^^ https://ja.wikipedia.org/wiki/%E6%A0%83%E3%83%8E%E5%BF%83%E5%89%9B%E5%8F%B2 栃ノ心剛史 (抜粋) 栃ノ心 剛史(つよし、1987年10月13日 - )は、ジョージア・ムツヘタ出身で春日野部屋所属の現役大相撲力士。本名はレヴァニ・ゴルガゼ(グルジア語表記: )。愛称はレヴァニ、角界のニコラス・ケイジ。身長192cm、体重177kg。得意技は右四つ、寄り、