[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 (658ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 http://rio2016.5ch.net/test/read.cgi/math/1514376850/

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56: １３２人目の素数さん [sage] 2017/12/29(金) 15:32:31.01 ID:UhtZ751q ガロアのスレ主がゴミクズ野郎なのがよく分かるな http://rio2016.5ch.net/test/read.cgi/math/1514376850/56

57: １３２人目の素数さん [] 2017/12/29(金) 17:19:18.60 ID:bDrChn34 スレ主は勉強するか黙るかどちらかにすべき http://rio2016.5ch.net/test/read.cgi/math/1514376850/57

58: １３２人目の素数さん [] 2017/12/29(金) 17:49:49.79 ID:41hqZa6e お兄さん「先生来られました」 おばあさん「先生？」 おじいさん「数学の先生」 俺「まあ…うん…数学やってる」 おばあさん「計算が得意なの？」 俺「計算は…あまり得意じゃない。計算するのが数学じゃなくて計算するための理論を組み立てるのが数学だから」 http://rio2016.5ch.net/test/read.cgi/math/1514376850/58

59: １３２人目の素数さん [sage] 2017/12/30(土) 12:02:04.90 ID:T5iI1wtu スレ主は今年も進歩ゼロでしたとさ http://rio2016.5ch.net/test/read.cgi/math/1514376850/59

60: １３２人目の素数さん [sage] 2017/12/30(土) 13:17:27.93 ID:dSbKeTYf ここまで酷いとホントにゴミだよ なんで数学板にいるの？ってレベル http://rio2016.5ch.net/test/read.cgi/math/1514376850/60

61: １３２人目の素数さん [] 2017/12/31(日) 11:42:22.49 ID:yDllqZzl >>58 これってスレ主？ http://rio2016.5ch.net/test/read.cgi/math/1514376850/61

62: １３２人目の素数さん [sage] 2017/12/31(日) 12:15:45.13 ID:65THZoXS ズレ主を表す今年の漢字2017は？ 「誤」「乱」「偽」「劣」「愚」 http://rio2016.5ch.net/test/read.cgi/math/1514376850/62

63: １３２人目の素数さん [] 2017/12/31(日) 15:12:52.22 ID:yDllqZzl スレ主の反応が無い。 冬休み、うつ、飽きた どれだ？ http://rio2016.5ch.net/test/read.cgi/math/1514376850/63

64: １３２人目の素数さん [] 2018/01/01(月) 01:44:15.87 ID:9ORABeV3 スレ主がこのまま消えますように http://rio2016.5ch.net/test/read.cgi/math/1514376850/64

65: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:04:59.09 ID:dCRrvhl7 皆さま、どうも。スレ主です。（＾＾ 明けまして、おめでとうございます。 新年も、よろしくお願いします。ｍ（＿ ＿）ｍ http://rio2016.5ch.net/test/read.cgi/math/1514376850/65

66: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:06:29.09 ID:dCRrvhl7 >>53 ID:gcYWyS10 さん、沢山レスありがとう 貴方のレスは、レベル高いね あとで、じっくり読むよ（＾＾ http://rio2016.5ch.net/test/read.cgi/math/1514376850/66

67: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:07:53.69 ID:dCRrvhl7 で、勝手ながら、年末年始に読んだ関連を貼るよ（＾＾ まず、関連参考：検索でヒットしたので貼る。 BaireCategory.pdfの”3. Pointwise limits of continuous functions.”に、「422に書いた定理」の関連記述 「Theorem. If f : R → R is a pointwise limit of continuous functions, then Df is Fσ meager (that is, a countable union of closed sets with empty interior). (In particular, by Baire's theorem, f is continuous

on a dense subset of R.)」とあり（当たり前か？ （＾＾ ） http://www.math.utk.edu/~freire/teaching/m447f16/m447f16index.html MATH 447- Advanced Calculus I- Fall 2016- A. FREIRE (or: ANALYSIS IN R^n) (抜粋) http://www.math.utk.edu/~freire/teaching/m447f16/BaireCategory.pdf Sets of discontinuity and Baire's theorem Baire Category Notes (5 problems) (the problems are HW8, due Friday 11/4)A. FREIRE 2016 (抜粋) 1. Sets of discontinuity. For f : R → R, we define Df = {x ∈ R; f is not continuous at xg: 3. Point

wise limits of continuous functions. Theorem. If f : R → R is a pointwise limit of continuous functions, then Df is Fσ meager (that is, a countable union of closed sets with empty interior). (In particular, by Baire's theorem, f is continuous on a dense subset of R.) Proof. We know Df = ∪ n>=1 D1/n (see Section 1), so it suffices to show that the closed sets Dε have empty interior, for any ε > 0. By contradiction, suppose Dε contains an open interval I. We'll find an open interval J ⊂ I disjoi

nt from Dε! Let fn → f pointwise on R, with each fn : R → R continuous. For each N >= 1, consider the set: CN = {x ∈ I; (∀m, n >= N)|fm(x) - fn(x)| <= ε/3}. Clearly ∪ N>=1 CN = I (by pointwise convergence). QED (引用終り) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/67

68: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:08:32.82 ID:dCRrvhl7 >>67 つづき （上記の関連参考：出典URL） http://www.math.utk.edu/~freire/teaching/m447f16/m447f16topics.html Math 447 Fall 2016- A. FREIRE TOPICS PART I: Topology Supplementary handouts (for advanced students): (adapted from more advanced classes and not yet in final form) http://www.math.utk.edu/~freire/teaching/m447f16/GeneralTopologyReview.pdf Definitions and Theorems from General Topology h

ttp://www.math.utk.edu/~freire/teaching/m447f16/BanachSpace.pdf Locally compact Banach spaces are finite dimensional (includes 4 problems) http://www.math.utk.edu/~freire/teaching/m447f16/SpacesOfContinuousFunctions.pdf Spaces of Continuous Functions (outdated) http://www.math.utk.edu/~freire/teaching/m447f16/StoneWeierstrassNotes.pdf Stone-Weierstrass theorem-notes (includes 6 problems) http://www.math.utk.edu/~freire/teaching/m447f16/AscoliArzelaNotes.pdf Ascoli-Arzela-Notes (final-included 7 exercises

with solutions, and 11 extra problems.) http://www.math.utk.edu/~freire/teaching/ Alex Freire Department of Mathematics University of Tennessee （終り） つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/68

69: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:10:25.74 ID:dCRrvhl7 >>68 つづき あと、いま、「422に書いた定理」に、似た文献を見つけて読んでいる。（＾＾ ”I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994” これ、出版されていて、アマゾンでもヒットした 疑問が二つ １）Proposition 1.1.1. の「Given ε > 0 there is a δ > 0 such that {x ∈ (x0 − δ, x0 + δ) : |f(x) − f(x0)| ≧ ε} ∈ J.」で、普通のεδ論法だと、 |f(x) − f(x0)| ＜

ε と不等号の向きが逆になると思うが、誤植か？ σ-ideal を考えているから、これで良いのか？ どうも良いみたいだが ２）Corollary 1.1.6. の「(ii): There exists a residual set K such that f|K is continuous.*2」で、f|Kは、Theorem 1.1.4.の”(ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous.”の記載ぶりとの比較から、f|Kcの誤記かなと思ったり？ 意味が全く違ってくる これにも、「422に書いた定理」の関連記述あり（後述） つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/69

70: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:10:50.85 ID:dCRrvhl7 >>69 つづき http://www.math.wvu.edu/~kcies/prepF/BookIdensity/BookIdensity.pdf I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994- 被引用数: 84 (抜粋) CHAPTER 1 The Ordinary Density Topology 1.1. A Simple Category Topology To gain some insight into what is happening with limits like this, it is useful to generalize this idea to a topological setting. A nonempty family J ⊂P(X) of subsets

of X is an ideal on X if A ⊂ B and B ∈ J imply that A ∈ J and if A∪B ∈ J provided A,B ∈ J. An ideal J on X is said to be a σ-ideal on X if ∪n∈N An ∈ J for every family {An : n ∈ N} ⊂ J. Let J be an ideal on R and Ｔo be the ordinary topology on R. The set T (J) = {G ＼ J : G ∈ Ｔo, J ∈ J} is a topology on R which is finer than Ｔo. The following proposition is evident from the definitions. Proposition 1.1.1. Let J be a σ-ideal on R and T (J ) be as above. For f : (R, T (J )) → (R, Ｔo) and

x0 ∈ R the following statements are equivalent to each other. (i): f is continuous at x0. (ii): Given ε > 0 there is a δ > 0 such that {x ∈ (x0 − δ, x0 + δ) : |f(x) − f(x0)| ≧ ε} ∈ J. つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/70

71: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:11:20.87 ID:dCRrvhl7 >>70 つづき Theorem 1.1.4. Let J be a σ-ideal and f : R → R. The following statements are equivalent. (i): The function f is J -continuous J -a.e. (ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous. Furthermore, if the ideal J contains no interval, then the following statement is equivalent to (i) and (ii) (iii): There exists a function g : R → R such that f

= g J -a.e. and g is continuous in the ordinary sense J -a.e. Proof. The fact that (ii) implies (i) is obvious. Suppose (i) is true and let f be J -continuous on a set M = Jc for J ∈ J. For each n ∈ N and x ∈ M, by Proposition 1.1.1(ii) there is an open interval I(n, x) and a J(n, x) ∈ J such that x ∈ I(n, x) \ J(n, x) ⊂ f−1((f(x) − 1/n, f(x) + 1/n)). For each fixed n, there must be a countable sequence xn,m ∈ M such that M ⊂∪m∈N I(n, xn,m). Let K = J ∪ ∪ n,m∈N J(n, xn,m) ∈ J. I

f x ∈ Kc and ε > 0, then there must exist natural numbers n and m such that 2/n < ε and x ∈ I(n, xn,m). Then |f(x) − f(xn,m)| < 1/n so that f(x) ∈ (f(xn,m) − 1/n, f(xn,m) + 1/n) ⊂ (f(x) − ε, f(x) + ε) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/71

72: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:11:49.95 ID:dCRrvhl7 >>71 つづき and I(n, xn,m) ∩ Kc ⊂ f−1((f(x) − ε, f(x) + ε)). Hence, f|Kc is continuous at x. To prove the last part of the theorem, note first that (iii) implies (ii) even without the restriction that J contains no interval. Now suppose that J contains no interval and that f,K are as in (ii). Define (1) G(x) = lim sup t→x,t∈Kc f(t) and (2) g(x) = G(x) when G(x) is finite, 　

or = f(x) otherwise. In particular, it follows from (ii) that f|Kc = g|Kc . Let x ∈ Kc and ε > 0. According to (ii) there is a δ > 0 such that (3) |g(y) − g(x)| = |f(y) − f(x)| < ε/2 whenever y ∈ (x − δ, x + δ) ∩Kc. If z ∈ (x − δ, x + δ) ∩K, then the assumption that K can contain no nonempty open set implies the existence of a sequence {zn : n ∈ N} ⊂ (x − δ, x + δ) ∩ Kc such that f(zn) → G(z). Hence, by (3), G(z) is finite, so g(z) = G(z) and |g(

z) − g(x)| ? ε/2 < ε. Therefore, g is continuous at x. QED つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/72

73: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:12:19.36 ID:dCRrvhl7 >>72 つづき The following example is interesting in light of the previous theorem. Example 1.1.5. Let I be the σ-ideal consisting of all first category subsets of R. I-continuity is often called qualitative continuity [26]. It is well-known in this case that f is a Baire function if, and only if, f is qualitatively continuous I-a.e. In particular, combining Example 1.1.5 with Theorem 1.1.4 yie

lds the following well-known corollary, which will be useful in the sequel. Corollary 1.1.6. Let f : R → R. The following statements are equivalent. (i): f is a Baire function. (ii): There exists a residual set K such that f|K is continuous.*2 (iii): f is qualitatively continuous I-a.e. In the case of Lebesgue measure, the following is true. *2 A set is residual if its complement is first category. This is often called comeager. つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/73

74: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:12:44.18 ID:dCRrvhl7 >>73 つづき If condition (i) in Theorem 1.1.4 is strengthened to everywhere, the following corollary results. Corollary 1.1.8. Let J be a σ-ideal which contains no nonempty open set. A function f : R → R is continuous everywhere if, and only if, it is J -continuous everywhere. Proof. If f is continuous, then it is clearly J -continuous. So, suppose f is J -continuous everywhere, x0 ∈ R and

ε > 0. Using Proposition 1.1.1(ii), there must be an ordinary open neighborhood G0 of x0 such that F0 = {x ∈ G0 : |f(x) − f(x0)| > ε} ∈ J. Suppose there is an x1 ∈ F0. Choose δ > 0 such that δ < |f(x1) − f(x0)| − ε. As before, there exists an ordinary open neighborhood G1 ⊂ G0 of x1 such that F1 = {x ∈ G1 : |f(x1) − f(x)| > δ} ∈ J. It is clear that G1 ⊂ F0 ∪ F1 ∈ J, because |f(x1) − f(x0)| > ε + δ. But, this implies J contains a nonempty

open set, which contradicts the condition placed on J in the statement of the corollary. This contradiction shows that F0 = Φ. The preceding corollary demonstrates that global J -continuity may not be a very useful concept. In particular, it is worthwhile noting for future reference that global I-continuity and global N-continuity are no different than ordinary continuity. (引用終り) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/74

75: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:13:26.56 ID:dCRrvhl7 sage http://rio2016.5ch.net/test/read.cgi/math/1514376850/75

76: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:13:34.44 ID:dCRrvhl7 >>74 つづき （上記の関連参考：出典URL） http://www.math.wvu.edu/~kcies/ Krzysztof Chris Ciesielski, Ph.D. Professor of Mathematics at Department of Mathematics, West Virginia University and Adjunct Professor at Medical Image Processing Group, Dept. of Radiology, Univ. of Pennsylvania. (抜粋) Books: (with L. Larson and K. Ostaszewski) I-density continuous functions, Memoirs of the AMS vol. 107 no

515, 1994; MR 94f:54035. (引用終り) https://www.amazon.co.jp/I-Density-Continuous-Functions-American-Mathematical/dp/0821825798 I-Density Continuous Functions (Memoirs of the American Mathematical Society) (英語) Krzysztof Ciesielski (著),? Lee Larson (著),? Krzysztof Ostaszewski (著) 1994/1/1 http://www.jstor.org/stable/44151978?seq=1#page_scan_tab_contents JOURNAL ARTICLE I-density Continuous Functions Krzysztof Ciesielski, Lee Larson and Krzysztof Ostaszewski Real Analysis Exchange Vol. 15, No. 1 (1989

-90), pp. 13-15 Published by: Michigan State University Press （終り） つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/76

77: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:14:10.20 ID:dCRrvhl7 >>76 つづき （参考：用語解説） https://en.wikipedia.org/wiki/Ideal_(set_theory) Ideal (set theory) (抜粋) In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of

any two elements of the ideal must also be in the ideal. More formally, given a set X, an ideal I on X is a nonempty subset of the powerset of X, such that: 1. Φ ∈ I 2.if A∈ I and B⊆ A, then B∈ I, and 3.if A,B∈ I, then A ∪ B∈ I Some authors add a third condition that X itself is not in I; ideals with this extra property are called proper ideals. Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exact

ly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set. Contents 1 Terminology 2 Examples of ideals 2.1 General examples 2.2 Ideals on the natural numbers 2.3 Ideals on the real numbers 2.4 Ideals on other sets 3 Operations on ideals 4 Relationships among ideals 5 See also 6 References (引用終り) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/77

78: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:14:45.77 ID:dCRrvhl7 >>77 つづき https://en.wikipedia.org/wiki/Sigma-ideal σ-ideal Sigma-ideal (Redirected from Σ-ideal) (抜粋) In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory. L

et (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if the following properties are satisfied: (i) O ∈ N; (ii) When A ∈ N and B ∈ Σ , B ⊆ A ⇒ B ∈ N; (iii) {A_n}_{n∈N }⊆ N→ ∪ _{n∈N }A_n∈ N. Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of σ-ideal is dual to that of a countably complete (σ-) filter. If a measure μ is given on (X,Σ), the set of μ-negligible se

ts (S ∈ Σ such that μ(S) = 0) is a σ-ideal. The notion can be generalized to preorders (P,?,0) with a bottom element 0 as follows: I is a σ-ideal of P just when (i') 0 ∈ I, (ii') x ? y & y ∈ I ⇒ x ∈ I, and (iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ? y for each n Thus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist). It is natural in this context to ask that P itself have countable suprema. A σ-ideal of a

set X is a σ-ideal of the power set of X. That is, when no σ-algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the σ-ideal generated by the collection of closed subsets with empty interior. (引用終り) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/78

79: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:15:30.82 ID:dCRrvhl7 >>78 つづき https://en.wikipedia.org/wiki/Ideal Ideal (抜粋) Mathematics Ideal (ring theory), special subsets of a ring considered in abstract algebra Ideal, special subsets of a semigroup Ideal (order theory), special kind of lower sets of an order Ideal (set theory), a collection of sets regarded as "small" or "negligible" Ideal (Lie algebra), a particular subset in a Lie al

gebra Ideal point, a boundary point in hyperbolic geometry Ideal triangle, a triangle in hyperbolic geometry whose vertices are ideal points (引用終り) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/79

80: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:16:04.01 ID:dCRrvhl7 >>79 つづき http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/maths/real_analysis_2009_abstract.pdf 典型的連続関数のDini微分（著者最終稿）斎藤新悟 実解析学シンポジウム2009報告集，pp. 25-33. （上記の関連参考：出典URL） http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/maths/papers.html 斎藤新悟 出版物 http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/ 斎藤新悟 九州大学基幹教育院准教授 1981年大阪府生まれ 東京大

学理学部数学科卒業 つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/80

81: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:16:33.36 ID:dCRrvhl7 >>80 つづき （以前のスレから関連抜粋） スレ46 https://rio2016.5ch.net/test/read.cgi/math/1510442940/398 ＜引用＞ http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, DIOPHANTINE APPROXIMATION, AND A REFORMULATION OF THE THUE-SIEGEL-ROTH THEOREM JUAN LUIS VARONA 2009 This paper has been published in Gazette of the Australian Mathem

atical Society, Volume 36, Number 5, November 2009, pp. 353{361. (抜粋) So, in this paper we are going to analyze the dierentiability of the real function fν(x) =0 if x ∈ R ＼ Q, 　　or =1/q^ν if x = p/q ∈ Q, irreducible, for various values of ν ∈ R. Theorem 1. For ν > 2, the function fν is discontinuous (and consequently not dierentiable) at the rationals, and continuous at the irrationals. With respect the dierentiability, we have: (a) For every irrational number x with bounded elements in its

continued fraction expansion, fν is differentiable at x. (b) There exist infinitely many irrational numbers x such that fν is not differentiable at x. Moreover, the sets of numbers that fulfill (a) and (b) are both of them uncountable. Theorem 2. For ν > 2, let us denote Cν = { x ∈ R : fν is continuous at x }, Dν = { x ∈ R : fν is dierentiable at x }. Then, the Lebesgue measure of the sets R ＼ Cν and R ＼ Dν is 0, but the four sets Cν, R ＼ Cν, Dν, and R ＼ Dν are dense in R. (引用終り)

つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/81

82: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:17:30.46 ID:dCRrvhl7 >>81 つづき で、”a nonempty open set”（ordinary open neighborhood ）が、結構重要キーワードじゃないかな？ R中のQのように稠密分散で、 R＼Qは、”a nonempty open set”の集まりになるけれども （似た状況は、上記の「the Lebesgue measure of the sets R ＼ Cν and R ＼ Dν is 0, but the four sets Cν, R ＼ Cν, Dν, and R ＼ Dν are dense in R.」とある通りで） 「422に書いた定理」の系1.8の背理法証明に使え

るような、区間(a, b)が取れると言えるかどうかだ？ 以上 http://rio2016.5ch.net/test/read.cgi/math/1514376850/82

83: １３２人目の素数さん [] 2018/01/01(月) 17:38:20.61 ID:WRx3yiBV >>82 >R中のQのように稠密分散で、 >R＼Qは、”a nonempty open set”の集まりになるけれども ？ http://rio2016.5ch.net/test/read.cgi/math/1514376850/83

84: １３２人目の素数さん [sage] 2018/01/01(月) 17:48:29.72 ID:HicRQN2S おっちゃんです。 今日は午前4時に散歩したら、新聞配達のお姉ちゃんが自転車で配達していた。 今は意識もうろうとしていて、もうお寝んねタイム。 http://rio2016.5ch.net/test/read.cgi/math/1514376850/84

85: １３２人目の素数さん [sage] 2018/01/01(月) 18:10:20.43 ID:HicRQN2S まあ、深夜に散歩するのも案外日常とは違う面白い光景が見られる。 深夜にコンビニに行く人も時々見かける。 昼間の車の排気ガスで汚れた空気とは違い、昼間程汚れていない新鮮な空気は吸えるな。 それじゃ、おっちゃん寝る。 http://rio2016.5ch.net/test/read.cgi/math/1514376850/85

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