数学の一分野、位相幾何学（いそうきかがく、英: topology, トポロジー[注釈 1]）は、その名称がギリシア語: τόπος（「位置」「場所」）と λόγος（「言葉」「学問」） に由来し、「位置の学問」を意味している。
歴史[編集]
オイラーによる「ケーニヒスベルクの 7 つの橋」問題の解決は位相幾何学の萌芽（のひとつ）であるとみなされている。
ユークリッド幾何学が紀元前にはできていたことと比較すると、オイラーやガウスに始まる位相幾何学は高々 250 年の歴史であり、大きな差がある。オイラーは、いわゆるオイラーの多面体定理において球面に連続的に変形できるような多面体の辺・頂点・面の数の間にある関係が成り立つことを見出したが、これをもって位相幾何学の始まりとするのが一般的である。
多面体の頂点、辺、面の数を各々 n0, n1, n2 とおくと、これらが n0 − n1 + n2 = 2 の関係にあるとするオイラーの定理は、18 世紀当時の解析学、代数学を中心とする数学の流れにおいて孤立した結果であった。19 世紀にガウスは絡み目数を線積分により表示する公式を与え、また後半紀にリーマンが現在リーマン面と呼ばれる概念を提出し、ロッホは曲面の上の 2 つの偏微分方程式の解の自由度の差を曲面の種数を含む数と同定するリーマン・ロッホの定理をまとめた。これら前駆的研究に対して、トポロジーがひとつの分野として確立する契機となったのは 1900 年前後のポワンカレの一連の研究による[5]。
現代的な位相幾何学は 19 世紀に後半に確立された集合論を大きな基盤として成り立っている。集合論の祖のひとりであるゲオルク・カントールはフーリエ級数の研究に際してユークリッド空間内の点集合について考察している。
カントール、ボルテラ、アルツェラ（英語版）、アダマール、アスコリ（英語版）、らの研究を取りまとめる形で（今日では一般的な位相空間の特別の場合であると考えられている）距離空間の概念を確立したのはフレシェで、1906 年のことである。「位相空間」という用語を導入したのはハウスドルフで、1914 年に今日ではハウスドルフ空間と呼ばれる概念を定義するために用いられたものであるが、その一般化として現代的な意味での位相空間という概念が確立されるのは 1922 年、クラトフスキー（英語版）の手による。
応用[編集]
![[icon]](https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Wiki_letter_w_cropped.svg){kind=small} |
この節の加筆が望まれています。
|
位相幾何学の手法を用いると、抽象的な接続関係に関する性質や微小変形で不変な大域的な性質を扱うことができる。数学の一分野として整理される以前より、位相幾何学的手法が単発的に使われてきた(空間中の二つの電流の相互作用に対する、ガウスの線積分表示など)が、二十世紀後半には特に他分野との関連が深まり、現在でも応用領域は広がっている。
| 応用領域 | 内容 |
|---|---|
| 物理学 | 宇宙の形状、素粒子の記述体系、量子数の記述、超伝導絶縁体、我々の世界に関する性質(タイムマシンは存在するか？など)。 |
| 物質科学 | フラーレンなど分子構造。 |
| 生命科学 | 結び目をなす分子の、形状による機能や変形(DNAトポイソメラーゼ)。 |
| 情報科学 | 論理体系の意味論を展開する枠組みとして層 (数学)の利用、経路空間のホモロジーによる記述。またネットワークの取り扱いにおいてはグラフ理論を手段として研究され、一般的にはネットワーク・トポロジーとして知られている。
また、人工知能の研究分野では「トポロジカル・データ・アナリシス」(Topological data analysis)技術が発展の見込みにある。
|
| カタストロフィー理論 | 形態形成、経済現象の記述。 |
| 3次元コンピュータグラフィックス | 3DCGにおけるモーフィングはホモトピー変形を利用している。また立体計測やデジタルスカルプトで生成された複雑なポリゴンモデルを単純な構造のモデルに作り変える操作をリトポロジー(Retopology)と呼ぶ。https://ja.wikipedia.org/wiki/%E4%BD%8D%E7%9B%B8%E5%B9%BE%E4%BD%95%E5%AD%A6
ゼロ除算の発見は日本です：
∞？？？
∞は定まった数ではない・
人工知能はゼロ除算ができるでしょうか：
とても興味深く読みました：
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
ゼロ除算関係論文・本
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\usepackage{color}
\usepackage{url}
%%%%%%% �}�ԍ��̃J�E���^
\newcounter{num}
\setcounter{num}{0}
%\setcounter{prop}{1}
\newcommand{\Fg}[1][]{\thenum}
\newcommand\Ra{r_{\rm A}}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 433:\\ Puha's Horn Torus Model for the Riemann Sphere From the Viewpoint of Division by Zero}
\author{
}
\date{2018.07.16}
\maketitle
\newcommand\Al{\alpha}
\newcommand\B{\beta}
\newcommand\De{\delta}
\def\z{\zeta}
\def\rA{r_{\rm A}}
{\bf Abstract: } In this announcement, we will introduce a beautiful horn torus model for the Riemann sphere in complex analysis from the viewpoint of the division by zero based on \cite{ps}.
\medskip
\section{Division by zero calculus and introduction}
The division by zero with mysterious and long history was indeed trivial and clear as in the followings:
\medskip
By the concept of the Moore-Penrose generalized solution of the fundamental equation $ax=b$, the division by zero was trivial and clear all as $a/0=0$ in the {\bf generalized fraction} that is defined by the generalized solution of the equation $ax=b$.
Division by zero is trivial and clear from the concept of repeated subtraction - H. Michiwaki.
Recall the uniqueness theorem by S. Takahasi on the division by zero.
The simple field structure containing division by zero was established by M. Yamada.
Many applications of the division by zero to Wasan geometry were given by H. Okumura.
\medskip
The division by zero opens a new world since Aristotelēs-Euclid.
See the references for recent related results.
As the number system containing the division by zero, the Yamada field structure is complete.
However, for applications of the division by zero to {\bf functions}, we need the concept of the division by zero calculus for the sake of uniquely determinations of the results and for other reasons.
For example, for the typical linear mapping
\begin{equation}
W = \frac{z - i}{z + i},
\end{equation}
it gives a conformal mapping on $\{{\bf C} \setminus \{-i\}\}$ onto $\{{\bf C} \setminus \{1\}\}$ in one to one and from \begin{equation}
W = 1 + \frac{-2i}{ z - (-i)},
\end{equation}
we see that $-i$ corresponds to $1$ and so the function maps the whole $\{{\bf C} \}$ onto $\{{\bf C} \}$ in one to one.
Meanwhile, note that for
\begin{equation}
W = (z - i) \cdot \frac{1}{z + i},
\end{equation}
we should not enter $z= -i$ in the way
\begin{equation}
[(z - i)]_{z =-i} \cdot \left[ \frac{1}{z + i}\right]_{z =-i} = (-2i) \cdot 0= 0 .
\end{equation}
\medskip
However, in many cases, the above two results will have practical meanings and so, we will need to consider many ways for the application of the division by zero and we will need to check the results obtained, in some practical viewpoints. We referred to this delicate problem with many examples.
Therefore, we will introduce the division by zero calculus. For any Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{-1} C_n (z - a)^n + C_0 + \sum_{n=1}^{\infty} C_n (z - a)^n,
\end{equation}
we obtain the identity, by the division by zero
\begin{equation}
f(a) = C_0.
\end{equation}
Note that here, there is no problem on any convergence of the expansion (1.5) at the point $z = a$, because all the terms $(z - a)^n$ are zero at $z=a$ for $n \ne 0$.
\medskip
For the correspondence (1.6) for the function $f(z)$, we will call it {\bf the division by zero calculus}. By considering the formal derivatives in (1.5), we {\bf can define any order derivatives of the function} $f$ at the singular point $a$; that is,
$$
f^{(n)}(a) = n! C_n.
$$
\medskip
{\bf Apart from the motivation, we define the division by zero calculus by (1.6).}
With this assumption, we can obtain many new results and new ideas. However, for this assumption we have to check the results obtained whether they are reasonable or not. By this idea, we can avoid any logical problems. -- In this point, the division by zero calculus may be considered as an axiom.
\medskip
For the fundamental function $W =1/ z $ we did not consider any value at the origin $z = 0$, because we did not consider the division by zero
$1/ 0$ in a good way. Many and many people consider its value by the limiting like $+\infty $ and $- \infty$ or the
point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or
based on the basic idea of Aristotle. --
For the related Greece philosophy, see \cite{a,b,c}. However, as the division by zero we will consider its value of
the function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely in
mathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidian geometry, analytic geometry, complex analysis and the theory of differential equations in an undergraduate level and furthermore to our basic ideas for the space and universe.
For the extended complex plane, we consider its stereographic projection mapping as the Riemann sphere and the point at infinity is realized as the north pole in the Alexsandroff's one point compactification.
The Riemann sphere model gives a beautiful and complete realization of the extended complex plane through the stereographic projection mapping and the mapping has beautiful properties like isogonal (equiangular) and circle to circle correspondence (circle transformation). Therefore, the Riemann sphere is a very classical concept \cite{ahlfors}.
Now, with the division by zero we have to admit the strong discontinuity at the point at infinity.
On this situation, V. Puha discovered the mapping of the extended complex plane to a beautiful horn torus at (2018.6.4.7:22) and its inverse at (2018.6.18.22:18).
Incidentally, independently of the division by zero, Wolfgang W. Daeumler has various special great ideas on horn torus as we see from his site:
\medskip
Horn Torus \& Physics ( https://www.horntorus.com/ ) 'Geometry Of Everything', intellectual game to reveal
engrams of dimensional thinking and proposal for a different approach to physical questions ...
\medskip
Indeed, Wolfgang Daeumler was presumably the first (1996) who came to the idea of the possibility of a mapping onto the horn torus. He expressed the idea of that on his private website (http://www.dorntorus.de). He was also, apparently, the first who to point out that zero and infinity are represented by one and the same point on the horn torus model of expanded complex plane.
\medskip
In this announcement, we will introduce simply the new horn torus model for the classical Riemann sphere from the viewpoint of the division by zero.
\section{Horn torus model}
We will consider the three circles stated by
$$
\xi^2 + \left(\zeta-\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^2,
$$
$$
\left(\xi-\frac{1}{4}\right)^2 + \left(\zeta-\frac{1}{2}\right)^2 = \left(\frac{1}{4}\right)^2,
$$
and
$$
\left(\xi+\frac{1}{4}\right)^2 + \left(\zeta-\frac{1}{2}\right)^2 = \left(\frac{1}{4}\right)^2.
$$
By rotation on the space $(\xi,\eta,\zeta)$ on the $(x,y)$ plane as in $\xi =x, \eta=y$ around $\zeta$ axis, we will consider the sphere with $1/2$ radius as the Riemann sphere and the horn torus made in the sphere.
The stereographic projection mapping from $(x,y)$ plane to the Riemann sphere is given by
$$
\xi = \frac{x}{x^2 + y^2 + 1},
$$
$$
\eta = \frac{y}{x^2 + y^2 + 1},
$$
and
$$
\zeta = \frac{x^2 + y^2}{x^2 + y^2 + 1}.
$$
The mapping from $(x,y)$ plane to the horn torus by Puha is given by
$$
\xi = \frac{2x\sqrt{x^2 + y^2}}{(x^2 + y^2 + 1)^2},
$$
$$
\eta = \frac{2y\sqrt{x^2 + y^2}}{(x^2 + y^2 + 1)^2},
$$
and
$$
\zeta = \frac{(x^2 + y^2 -1)\sqrt{x^2 + y^2}}{(x^2 + y^2 + 1)^2} + \frac{1}{2}.
$$
The inversion is given by
$$
x = \xi \left(\xi^2 + \eta^2 + \left(\zeta - \frac{1}{2} \right)^2 -\zeta + \frac{1}{2} \right)^{(-1/2)}
$$
and
$$
y = \eta \left(\xi^2 + \eta^2 + \left(\zeta - \frac{1}{2} \right)^2 -\zeta + \frac{1}{2} \right)^{(-1/2)}.
$$
\section{Properties of horn torus model}
At first, the model shows the strong symmetry of the domains $\{|z|<1\}$ and $\{|z|>1\}$ and they correspond to the lower part and the upper part of the horn torus, respectively. The unit circle $\{|z|=1\}$ corresponds to the circle
$$
\xi^2 + \eta^2 = \left(\frac{1}{2}\right)^2, \quad \zeta = \frac{1}{2}
$$
in one to one way. Of course, the origin and the point at infinity are the same point and correspond to $(0,0,1/2)$. Furthermore,
the inversion relation
$$
z \longleftrightarrow \frac{1}{\overline{z}}
$$
with respect to the unit circle $\{|z|=1\}$ corresponds to the relation
$$
(\xi,\eta,\zeta) \longleftrightarrow (\xi,\eta, 1-\zeta)
$$
and similarly,
$$
z \longleftrightarrow -z
$$
corresponds to the relation
$$
(\xi,\eta,\zeta) \longleftrightarrow (- \xi,-\eta, \zeta)
$$
and
$$
z \longleftrightarrow - \frac{1}{\overline{z}}
$$
corresponds to the relation
$$
(\xi,\eta,\zeta) \longleftrightarrow (-\xi,-\eta, 1-\zeta)
$$
(H.G.W. Begehr: 2018.6.18.19:20).
Furthermore, we can see directly the important properties that the mapping is isogonal (equiangular) and infinitely small circles correspond
to infinitely small circles, as in analytic functions. However, of course, circles to circles mapping property is, in general, not valid as in the case of the stereographic projection mapping.
Horn torus, in contrast to the Riemann sphere, does not satisfy the definition of simply connected space because a closed nonzero path passing through the point $(0,0,1/2)$ can not be continuously shrinked to the point. In particular, note that a curve can pass the point $(0,0,1/2)$ on the horn torus.
We note that only zero and numbers of the form $|a|=1$ have the property : $ |a|^b=|a|, b\ne 0.$
Here, note that we can also consider $0^b =0$ (\cite{mms18}). The symmetry of the horn torus model agrees perfectly with this fact. Only zero and numbers of the form $|a|=1$ correspond to points on the plane described by equation $\zeta -1/2=0$. Only zero and numbers of the form $|a| =1$ correspond to points whose tangent lines to the surface of the horn torus are parallel to the axis $\zeta$.
\section{Conclusion}
The division by zero shows the strong discontinuity at the point at infinity, however, the Riemann sphere model and stereographic projection mapping are fundamental and beautiful.
Many people feel strange feelings for the strong discontinuity that is introduced by the division by zero to the Riemann sphere, however, the strong discontinuity appears in the universe naturally as we see from our new and many concrete results since Euclid.
However, the beautiful horn torus model may be accepted with great pleasures as our space idea. In particular, note that the domains $\{|z|<1\}$ and $\{|z|>1\}$ are completely conformally equivalent and so the completely symmetric property of the corresponding domains on the horn torus is very fine and from this viewpoint, the Riemann sphere model will be curious, in particular, at the point at infinity and the point at infinity will be vague.
\section{Acknowledgements}
The Insitute of Reproducing Kernels wishes to express its deep thanks Professors and colleagues H.G.W. Begehr, Wolfgang W. Daeumler, Hiroshi Okumura, Vyacheslav Puha and Tao Qian for their exciting communications.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms16}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$,
Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.2016.62007.
\bibitem{mms18}
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 293-305.
\bibitem{msy}
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 1-16.
\bibitem{osm}
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.
\bibitem{os}
H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).
\bibitem{o}
H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.
\bibitem{os18april}
H. Okumura and S. Saitoh,
Harmonic Mean and Division by Zero,
Dedicated to Professor Josip Peˇ cari´c on the occasion of his 70th birthday, Forum Geometricorum, {\bf 18} (2018), 155—159.
\bibitem{os18}
H. Okumura and S. Saitoh,
Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.
\bibitem{os18e}
H. Okumura and S. Saitoh,
Applications of the division by zero calculus to Wasan geometry.
GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES” (GJARCMG)(in press).
\bibitem{ps18}
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 399-418.
\bibitem{ps}
V. Puha and S. Saitoh,
Horn Torus Model for the Riemann Sphere From the Viewpoint of Division by Zero (manuscript).
\bibitem{s14}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{s16}
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer)
\bibitem{s17}
S. Saitoh, Mysterious Properties of the Point at Infinity, arXiv:1712.09467 [math.GM](2017.12.17).
\bibitem{s18}
S. Saitoh, Division by zero calculus (189 pages): http//okmr.yamatoblog.net/
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{a}
https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf
\bibitem{b}
http://publish.uwo.ca/~jbell/The 20Continuous.pdf
\bibitem{c}
http://www.mathpages.com/home/kmath526/kmath526.htm
\end{thebibliography}
\end{document}
                    |
0 件のコメント:
コメントを投稿