[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）18 (1002ﾚｽ)
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580: 08/04(日)07:20 ID:MRMarsEu(1/17) AAS
>>576
>これ比較的分かり易いね
>>578
>”R^2, R^^2の代わりに複素平面C及びRiemann球面C^”
>を使うのは、あざやかで分かり易いね
何をどうわかったんだか
>さすがのガウスさんも 一見単純なJordanの曲線に
>こんなにネチッコイ 位相空間の議論があるとは、夢にも思わなかったかでしょう
>ガウスさんの後世に、数学で病的な例がいろいろ発見された歴史がありますから
「実数の無限列で、各項が正の値なら、∞に発散する」
省1
581: 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)08:35 ID:oj4WjR/C(1/17) AAS
>>578 補足
>ジョルダン-シェーンフリースの定理（英語版）を参照されたい
下記ですね
https://en.wikipedia.org/wiki/Jordan_curve_theorem
Jordan curve theorem
Proof and generalizations
There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that the interior and the exterior planar regions determined by a Jordan curve in R2 are homeomorphic to the interior and exterior of the unit disk. In particular, for any point P in the interior region and a point A on the Jordan curve, there exists a Jordan arc connecting P with A and, with the exception of the endpoint A, completely lying in the interior region. An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve φ: S1 → R2, where S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2 → R2 of the plane. Unlike Lebesgue's and Brouwer's generalization of the Jordan curve theorem, this statement becomes false in higher dimensions: while the exterior of the unit ball in R3 is simply connected, because it retracts onto the unit sphere, the Alexander horned sphere is a subset of R3 homeomorphic to a sphere, but so twisted in space that the unbounded component of its complement in R3 is not simply connected, and hence not homeomorphic to the exterior of the unit ball.
https://en.wikipedia.org/wiki/Schoenflies_problem
Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
省7
582(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)08:36 ID:oj4WjR/C(2/17) AAS
つづき
Proofs of the Jordan–Schoenflies theorem
For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way.
Polygonal curve
Continuous curve
Smooth curve
Proofs in the smooth case depend on finding a diffeomorphism between the interior/exterior of the curve and the closed unit disk (or its complement in the extended plane). This can be solved for example by using the smooth Riemann mapping theorem, for which a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels.[10]
省8
583: 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)08:43 ID:oj4WjR/C(3/17) AAS
>>582
(引用開始)
Smooth curve
Proofs in the smooth case depend on finding a diffeomorphism between the interior/exterior of the curve and the closed unit disk (or its complement in the extended plane). This can be solved for example by using the smooth Riemann mapping theorem, for which a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels.[10]
(引用終り)
あら、こんなところに Bergman kernelが
面白いね
(参考)
https://en.wikipedia.org/wiki/Bergman_kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic functions on a domain D in Cn.
省6
584: 08/04(日)09:36 ID:MRMarsEu(2/17) AAS
また理解もできないことをコピペしてドヤる病気が再発しちゃいましたか
無能な人が嘘ついてまで有能だと自慢するって、完全に病んでますね
585(12): 08/04(日)09:41 ID:MRMarsEu(3/17) AAS
アレクサンダーの角付き球面（Alexander horned sphere）は、
1924年にジェームズ・ワデル・アレクサンダー2世によって発見された、
トポロジーにおける病的な対象である。
ジョルダン曲線定理を拡張したジョルダン–シェーンフリースの定理、
それを更に高次元へと拡張した主張
「n 次元空間 Rn に埋め込まれた (n − 1) 次元球面 S(n − 1) に対し，
Rn − S(n − 1) の有界な連結成分の閉包は n 次元単位球とアイソトピックである．」
に対する3次元 (n = 3) における反例
（アレクサンダーの角付き球面の外部の領域の閉包は3次元球とならない）
として知られている。
586(1): 08/04(日)09:43 ID:MRMarsEu(4/17) AAS
>>585 構成も奇妙さもなんかキャッソン・ハンドルに似てる気がするのは気のせいか？
587(1): 08/04(日)09:50 ID:u61j/16w(1) AAS
ベルグマン核は一変数関数論でも重要
588: 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)11:30 ID:oj4WjR/C(4/17) AAS
>>585
おサルさん>>5 さ
君は、倒錯している
その文は、ウィキペディア https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%AC%E3%82%AF%E3%82%B5%E3%83%B3%E3%83%80%E3%83%BC%E3%81%AE%E8%A7%92%E4%BB%98%E3%81%8D%E7%90%83%E9%9D%A2
からの盗用だよ
それ、犯罪ですよ
一方、出典と著者それにURLを明示して文章を引用するのは可
盗用ではありません
おサルさん
君は、倒錯している
589(1): 08/04(日)11:37 ID:MRMarsEu(5/17) AAS
犯罪者が犯罪を告発
まず自分を処刑せよ
話はその後だ
自ら首をはねよ ニッポンジン!
590: 08/04(日)11:39 ID:MRMarsEu(6/17) AAS
なんなら互いに首をはねあうか
貴様が死ぬなら俺も死ぬぞ
ともに地獄に堕ちようぞ
591: 08/04(日)11:41 ID:MRMarsEu(7/17) AAS
さぁ、左から右へ、一気に掻き切れ！
592(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)12:54 ID:oj4WjR/C(5/17) AAS
>>586
>>>585 構成も奇妙さもなんかキャッソン・ハンドルに似てる気がするのは気のせいか？
気のせいではないかもしれん
Alexander horned sphereについて、R. H. Bing の1952年の仕事がある（下記）
R. H. Bingは、ポアンカレ予想に取り組んでいた
Cassonもまた、3- and 4-dimensional topologyに取り組んでいたらしい
Bingの影響を受けている気がする
(参考)
https://en.wikipedia.org/wiki/Alexander_horned_sphere
Alexander horned sphere
省7
593: 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)12:55 ID:oj4WjR/C(6/17) AAS
つづき
https://en.wikipedia.org/wiki/R._H._Bing
R. H. Bing (October 20, 1914 – April 28, 1986)
Mathematical contributions
In 1951, he proved results regarding the metrizability of topological spaces, including what would later be called the Bing–Nagata–Smirnov metrization theorem.
In 1952, Bing showed that the double of a solid Alexander horned sphere was the 3-sphere. This showed the existence of an involution on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere, which meant that the original Smith conjecture needed to be phrased in a suitable category. This result also jump-started research into crumpled cubes. The proof involved a method later developed by Bing and others into set of techniques called Bing shrinking. Proofs of the generalized Schoenflies conjecture and the double suspension theorem relied on Bing-type shrinking.
Bing was fascinated by the Poincaré conjecture and made several major attacks which ended unsuccessfully, contributing to the reputation of the conjecture as a very difficult one. He did show that a simply connected, closed 3-manifold with the property that every loop was contained in a 3-ball is homeomorphic to the 3-sphere.
Bing was responsible for initiating research into the Property P conjecture, as well as its name, as a potentially more tractable version of the Poincaré conjecture. It was proven in 2004 as a culmination of work from several areas of mathematics. With some irony, this proof was announced some time after Grigori Perelman announced his proof of the Poincaré conjecture.
つづく
594: 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)12:56 ID:oj4WjR/C(7/17) AAS
つづき
https://en.wikipedia.org/wiki/Andrew_Casson
Andrew John Casson FRS (born 1943) is a mathematician, studying geometric topology. Casson is the Philip Schuyler Beebe Professor of Mathematics[1] at Yale University.
Work
Casson has worked in both high-dimensional manifold topology and 3- and 4-dimensional topology, using both geometric and algebraic techniques.
Among other discoveries, he contributed to the disproof of the manifold Hauptvermutung, introduced the Casson invariant, a modern invariant for 3-manifolds, and Casson handles, used in Michael Freedman's proof of the 4-dimensional Poincaré conjecture.
https://en.wikipedia.org/wiki/Casson_handle
In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure.
Motivation
In the proof of the h-cobordism theorem, the following construction is used. Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold whose boundary is the given circle. If the manifold is simply connected then we can find a map from a disc to the manifold with boundary the given circle, and if the manifold is of dimension at least 5 then by putting this disc in "general position" it becomes an embedding. The number 5 appears for the following reason: submanifolds of dimension m and n in general position do not intersect provided the dimension of the manifold containing them has dimension greater than m+n.
省2
595: 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)12:59 ID:oj4WjR/C(8/17) AAS
つづき
If the manifold is 4 dimensional, this does not work: the problem is that a disc in general position may have double points where two points of the disc have the same image. This is the main reason why the usual proof of the h-cobordism theorem only works for cobordisms whose boundary has dimension at least 5. We can try to get rid of these double points as follows. Draw a line on the disc joining two points with the same image. If the image of this line is the boundary of an embedded disc (called a Whitney disc), then it is easy to remove the double point. However this argument seems to be going round in circles: in order to eliminate a double point of the first disc, we need to construct a second embedded disc, whose construction involves exactly the same problem of eliminating double points.
Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit.
en.wikipedia.org/wiki/Nagata%E2%80%93Smirnov_metrization_theorem
Nagata–Smirnov metrization theorem
ja.wikipedia.org/wiki/%E9%95%B7%E7%94%B0%E6%BD%A4%E4%B8%80
長田 潤一（ながた じゅんいち、1925年 - 2008年11月6日 ）は日本の数学者。専門は一般位相空間論。
森田紀一の指導を受ける。テキサスクリスチャン大学、ピッツバーグ大学、アムステルダム大学、大阪市立大学、大阪教育大学教授。1950年に位相空間が距離化可能であるための必要十分条件を与える長田-スミルノフの距離化定理を証明した。
(引用終り)
以上
596: 08/04(日)13:54 ID:MRMarsEu(8/17) AAS
>>592
>>構成も奇妙さもなんかキャッソン・ハンドルに似てる気がするのは気のせいか？
> 気のせいではないかもしれん
そうだとしても、高校卒業で数学諦めた君には全然関係ない話だったな
ま、貴様も首刎ねられて死ぬ勇気はなかったか
なら、数学板に書くのはやめて碁でも打ってな チキン🐓
597(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)14:13 ID:oj4WjR/C(9/17) AAS
>>587
>ベルグマン核は一変数関数論でも重要
ふむふむ
貼っておきますね
https://www.jstage.jst.go.jp/article/sugaku1947/48/4/48_4_415/_pdf/-char/ja
数学 1996 Volume 48 Issue 4 Pages 415-418
日本数学会50周年記念企画
多変数関数論の成立から一つの展望まで 大沢 健夫
岡 とは独立に,Bremerman[3],Norguet[11]もLevi問題を解いてはいるが,独自の影響力を持つにはいたらなかつた.
一方,一変数関数論の重要な主題である等角写像の理論は,
省16
598(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 08/04(日)14:17 ID:oj4WjR/C(10/17) AAS
>>589
ずばり
「盗人猛猛しい」
（とがめられても居直ったり、くってかかるさま）
だな
https://kotobank.jp/word/%E7%9B%97%E4%BA%BA%E7%8C%9B%E3%80%85%E3%81%97%E3%81%84-2236331
コトバンク
ことわざを知る辞典 「盗人猛猛しい」の解説
盗人猛々しい
悪事を働いて、とがめられても平然としているさま。
省1
599(1): 08/04(日)15:19 ID:MRMarsEu(9/17) AAS
>>598
>「盗人猛猛しい」
再三のコピペをとがめられて脊髄反射でキーキー吼えた大阪の🐒のことですな
🐒は数学板では駆除の対象 悪く思うな 次は人間に生まれることだな
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