ジョルジュ・ルメートル　生誕124周年

この日、「ベルギー」をはじめとして、いくつかの国で、Google検索のロゴが以下のものに変更されています。

124e verjaardag van Georges Lemaître

Georges Lemaître’s 124th Birthday

124 de ani de la nașterea lui Georges Lemaître

ジョルジュ・ルメートル 生誕124周年

ロゴ大

ロゴ小

表示された国

この日、「Google」のロゴが変更された国は以下の画像の通りです。

「ベルギー」をはじめとして、いくつかの国で表示されています。

「ジョルジュ・ルメートル」さんって？？

Georges-Henri Lemaître：ジョルジュ＝アンリ・ルメートル

ベルギー出身の天文学者です。

「ビッグバン」理論の提唱者として知られています。

「１８９４年７月１７日」に生まれて、「１９６６年６月２０日」に逝去されています。

そのため、生誕１２４周年となります。

ベルギー南部の「シャルルロワ」で生まれます。

イエズス会の学校で人文科学を学んだ後、１７歳でルーヴェン・カトリック大学の土木工学科に入学。

第一次世界大戦時には、勉強を中断し、ベルギー陸軍に志願入隊し従軍しています。

戦後、復学し、数学の分野で博士号を取得。

敬虔なクリスチャンであり、「カトリック司祭」として叙階もされましたが、引き続き勉学をすることを選択。

１９２３年、ケンブリッジ大学で、「エディントン光度」で知られる、天文学者のアーサー・エディントンから天文学を学び、翌年（１９２４年）には、ハーバード大学天文台で研究を行いました。

１９２５年、ベルギーへと戻り、母校のルーヴェン・カトリック大学の講師となりました。

１９２７年、１つの論文を発表します。

「Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extragalactiques：銀河系外星雲の視線速度を説明する、一定質量で半径が成長する宇宙」

この論文をきっかけに、彼は膨張宇宙論を展開します。

「宇宙は原子爆発から始まった」というビッグバン理論を提唱。

当時は「宇宙の始まり」などを提唱する学者は皆無でしたので、彼の考えは物議を呼びました。

彼の理論は、アインシュタインの相対性理論をベースに構築されたものでしたが、当のアインシュタイン本人を含め、著名な数学者、物理学者、天文学者は、彼の理論に懐疑的でした。

というのも、彼が司祭の叙階を受けており、「科学での宇宙論」と、「宗教上の宇宙論」が混同したものであるという印象を与えたからでした。

彼自身は、科学と宗教を混同することなく研究を行いましたが、周りでは印象による誤解、偏見があり、「ビッグバン」＝「聖書における天地創造」のような感じで捉えられました。

その後、いくつか論文を発表、彼自身も学会で詳しく説明することで、彼の理論は少しずつ認められるようになりました。

１９２９年、「エドウィン・ハッブル」さんと「ミルトン・ヒューメイソン」さんによって「ハッブルの法則」が発表されました。

この法則の発見が、宇宙は膨張しているという彼の理論の正しさを裏付けることとなります。

１９３４年にはベルギーの科学分野の賞である「フランキ賞」を受賞。

１９５３年には、イギリス王立天文学会が、天文学の分野で業績のあった研究者に贈る賞「エディントン・メダル」を設立、その最初の受賞者となりました。

宇宙創生の「ビッグバン理論」を提唱した彼については、同じ論理を提唱した「エドウィン・ハッブル」さんや、「ジョージ・ガモフ」さんと比べて、あまり知られていません。

それは、彼が司祭であったため、名声を求めていなかったためではないかと言われています。https://www.googletop.info/?p=442005

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

https://ameblo.jp/syoshinoris/entry-12370797278.html

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\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

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{\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.

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\end{document}