現代数学の系譜工学物理雑談古典ガロア理論も読む49

[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)
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ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

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(1/27) AAS
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

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(2/27) AAS
>>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（＾＾

67(1): 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(3/27) AAS
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. Pointwise 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 disjoint 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

68(1): 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(4/27) AAS
>>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

http://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

69(1): 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(5/27) AAS
>>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

70(1): 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(6/27) AAS
>>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

71(1): 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(7/27) AAS
>>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.
If 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

72(1): 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(8/27) AAS
>>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

73(1): 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(9/27) AAS
>>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 yields 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

74(1): 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(10/27) AAS
>>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

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(11/27) AAS
sage

76(1): 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(12/27) AAS
>>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

77(1): 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(13/27) AAS
>>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 exactly 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

78(1): 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(14/27) AAS
>>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.

Let (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 sets (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

79(1): 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(15/27) AAS
>>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 algebra
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

80(1): 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(16/27) AAS
>>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

81(2): 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(17/27) AAS
>>80 つづき

（以前のスレから関連抜粋）
スレ46 2chｽﾚ:math
＜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 Mathematical 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

82(2): 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(18/27) AAS
>>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

86(1): 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) 19:50:40.49 ID:dCRrvhl7(19/27) AAS
Thomae（「ポップコーン」）関数の絵が面白いので、ご紹介。
https://arxiv.org/abs/1702.06757
https://arxiv.org/pdf/1702.06757
Number-theoretic aspects of 1D localization: "popcorn function" with Lifshitz tails and its continuous approximation by the Dedekind eta S. Nechaev, K. Polovnikov (Submitted on 22 Feb 2017 (v1), last revised 26 Feb 2017 (this version, v2))
(抜粋)
We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers.

The spectral density of ensemble of linear polymer chains distributed with the law ?fL (0<f<1),

where L is the chain length, serves as a particular example.

At f→1, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function.

We suggest a continuous approximation of the popcorn function, based on the Dedekind η-function near the real axis.

Moreover, we provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at the spectral edges.

We emphasize that the ultrametric structure of the spectral density is ultimately connected with number-theoretic relations on asymptotic modular functions.

We also pay attention to connection of the Dedekind η-function near the real axis to invariant measures of some continued fractions studied by Borwein and Borwein in 1993.
(引用終り)

88: 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) 20:35:10.81 ID:dCRrvhl7(20/27) AAS
>>86

Figure 5: Plots of everywhere continuous f1(x) = -ln |η(x + iε)| (blue) and discrete f2(x) = Π/(12ε) g^2(x) (red) for ε = 10^-6 at rational points in 0 < x < 1.
が面白いね

89: 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) 20:36:14.06 ID:dCRrvhl7(21/27) AAS
>>84-85
おっちゃん、どうも、スレ主です。
レスありがとう
今年もよろしく（＾＾

90(4): 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) 21:08:33.72 ID:dCRrvhl7(22/27) AAS
（追加貼付）
スレ47 2chｽﾚ:math
245 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE [sage] 投稿日：2017/12/02
ちょっと、ピエロの過去レス46に戻る

スレ46 2chｽﾚ:math
（ピエロ）
181 返信：１３２人目の素数さん[sage] 投稿日：2017/11/15(水) 19:42:29.78 ID:fz0TcIh0 [2/3]
(抜粋)
さらにいえば、1/q^nを1/e^(-q)に置き換えても
リュービル数では微分不可能https://kbeanland.files.wordpress.com/2010/01/beanlandrobstevensonmonthly.pdf
(引用終り)

これ、結構面白ね（＾＾
要するに、Proposition 3.1で、無理数で０で有理数でプラス（T(x)＞０ xは有理数）となるどんな関数も、必ずどこか微分不可能な無理数があり、それは稠密だというのだ（下記PDF）

https://kbeanland.wordpress.com/research-articles/
Kevin Beanland ASSOCIATE PROFESSOR OF MATHEMATICS in the Department of Mathematics at Washington and Lee University.

Research Articles
My main research area is Banach space theory but, I have some work in real analysis and know some descriptive set theory as it applies to Banach space theory.

https://kbeanland.files.wordpress.com/2010/01/beanlandrobstevensonmonthly.pdf
Modifications of Thomae’s function and differentiability, (with James Roberts and Craig Stevenson) Amer. Math. Monthly, 116 (2009), no. 6, 531-535.
(抜粋)
3. A DENSE SET. While attempting to prove that T(1/n2) is differentiable on the irrationals,
we discovered that quite the opposite is actually true. In fact, as the following
proposition indicates, functions that are zero on the irrationals and positive on the rationals
will always be non-differentiable on a rather large set.

Proposition 3.1. Let f be a function on R that is positive on the rationals and 0 on
the irrationals. Then there is an uncountable dense set of irrationals on which f is not
differentiable.
(引用終り)

91(1): 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) 21:12:55.81 ID:dCRrvhl7(23/27) AAS
>>51
C++さん、どうも。スレ主です。
年末は、ばたばたして、お相手できませんでしたが

新年おめでとうございます
今年もよろしくお願いします。ｍ（＿＿）ｍ

92: 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) 23:33:24.50 ID:dCRrvhl7(24/27) AAS
Liouville Numbers について、調べていたら、下記ヒット
http://www.mathematik.uni-wuerzburg.de/~steuding/elaz2014.pdf
On Liouville Numbers - Yet Another Application of Functional Analysis To Number Theory Vortrag auf der ELAZ 2014 in Hildesheim: Jorn Steuding
(抜粋)
P21/42
Category vs. Measure
The set
L = (R ＼ Q) ∩ n>=1 (∪q>=2 ∪p (p/q -1/q^n ,p/q+1/q^n ))
of Liouville numbers is
・ big in the sense of category
(residual, dense Gδ),
・ small in the sense of measure
(Lebesgue measure zero, Hausdorff measure zero).
For the set of normal numbers it is the other way around.
(引用終り)

http://www.mathematik.uni-wuerzburg.de/~steuding/
Prof. Dr. Jorn Steuding Universitat Wurzburg Institut fur Mathematik

93(1): 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) 23:34:23.48 ID:dCRrvhl7(25/27) AAS
（参考）
https://ja.wikipedia.org/wiki/%E3%83%AA%E3%82%A6%E3%83%B4%E3%82%A3%E3%83%AB%E6%95%B0
リウヴィル数

https://en.wikipedia.org/wiki/Liouville_number
(抜粋)
Structure of the set of Liouville numbers[edit]
For each positive integer n, set

U_n=∪_q=2〜∞ ∪_p= -∞ 〜∞ {x∈ R :0<|x - p/q|< 1/q^n｝ =∪_q=2〜∞ ∪_p= -∞〜∞ ( p/q - 1/q^n, p/q+ 1/q^n)＼ { p/q}

The set of all Liouville numbers can thus be written as

L=∩_n=1〜∞ U_n.

Each Un is an open set; as its closure contains all rationals (the p/q's from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a dense Gδ set.

Along with the above remarks about measure, it shows that the set of Liouville numbers and its complement decompose the reals into two sets, one of which is meagre, and the other of Lebesgue measure zero.

つづく

94(1): 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) 23:35:35.73 ID:dCRrvhl7(26/27) AAS
>>93 つづき
Irrationality measure

The irrationality measure (or irrationality exponent or approximation exponent or Liouville?Roth constant) of a real number x is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any n in the power of q, we find the least upper bound of the set of real numbers μ such that

0<|x - p/q|< {1/q^μ

is satisfied by an infinite number of integer pairs (p, q) with q > 0. This least upper bound is defined to be the irrationality measure of x.[3]:246 For any value μ less than this upper bound, the infinite set of all rationals p/q satisfying the above inequality yield an approximation of x.
Conversely, if μ is greater than the upper bound, then there are at most finitely many (p, q) with q > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of q. In other words, given the irrationality measure μ of a real number x, whenever a rational approximation x ? p/q, p,q ∈ N yields n + 1 exact decimal digits, we have

1/10^n >= |x - p/q| >= {1/q^(μ +ε)

for any ε>0 except for at most a finite number of "lucky" pairs (p, q).

For a rational number α the irrationality measure is μ(α) = 1.[3]:246 The Thue?Siegel?Roth theorem states that if α is an algebraic number, real but not rational, then μ(α) = 2.[3]:248

Almost all numbers have an irrationality measure equal to 2.[3]:246

Transcendental numbers have irrationality measure 2 or greater. For example, the transcendental number e has μ(e) = 2.[3]:185 The irrationality measure of π is at most 7.60630853: μ(log 2)<3.57455391 and μ(log 3)<5.125.[4]

The Liouville numbers are precisely those numbers having infinite irrationality measure.[3]:248
(引用終り)

95(2): 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) 23:43:31.65 ID:dCRrvhl7(27/27) AAS
>>83
>>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
>？

リウヴィル数をイメージしてもらえば、良いのでは？稠密分散で、”a nonempty open set”の集まり
例えば
Structure of the set of Liouville numbers より
”Each Un is an open set; as its closure contains all rationals (the p/q's from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a dense Gδ set.”

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